When you mulch mow you need to be careful and not go too fast and leave a trail of visible clippings behind because those clipping on top of the grass can smother and they are out of reach of the Soil Food Web. If your forward speed is very slow and you still leave clippings on top then you need to cut more often and/or higher.
I will try to wait until mid October before feeding my lawn so I do not induce a growth spurt in the fall that might not be hardened off before a freeze, something not a problem for you, normally.
Check with the people at your counties University of Florida Cooperative Extension Service office about what is a good height to cut your grasses.

Grass clippings tend to stay on top of St. Augustine and Zoysia. I'd bag them and use them around trees and shrubs as mulch. Continue using Milorganite a couple times a year. Very effective. Ironite is also useful, but you only need half the recommended amount twice as often. Florida soils drain so fast the iron won't stay near the surface long enough to help the grass.

A little info about the shrubs and trees in your zoysia and St. Augustine would be helpful. Both love the constant sunshine, and both are native to China. Compost tea and top dressing with peat moss can be useful for either. You can get cubes of peat moss at a number of stores from Ace to Lowes/Home Depot to Tractor Supply. I'd get some white clover seed and thinly apply it in the spring if you don't mind clover in your grass. It will keep nitrogen near the surface and both of those grasses love nitrogen.

Key in Florida is your stormwater. Try and go Florida Friendly (see UFAS) near any sidewalk or street. Use mondo grass plugs, and they will grow very fast down there. Mulch heavily in between the mondo grass. Follow Florida Friendly guidelines. You want to keep moisture from the streets and storm sewers and in your yard feeding your plants. In areas that pool water, get some ginger at the local Publix, carve it up into 2-inch sections and shove it into the ground about six inches deep. It will grow a beautiful ginger lilies and repel bugs, bufo toads, Cuban tree frogs from that vicinity. Dwarf coffee makes a great shade bush if you have some shady areas.

Thatch is your biggest enemy in both of those grasses. Zoysia is the worst, and St. Augustine is the most sensitive to bugs. If you get too much thatch, sod webworm WILL kill your St. Augustine in the blink of an eye. Some folks like to mow it high, but consistent mowing at 2-inches is perfect for a healthy St. Augustine or Zoysia lawn. The latter can be cut much lower. The higher you let it grow, the higher the thatch layer and the more likely future problems. You don't have to worry about deep roots on these grasses in your zone because rain usually isn't an issue, and nutrients don't hang around the top layer of soil for long. They flow right through the sandy soil. So don't bother buying any inorganic fertilizer unless you just need an instant green-up, but hey, Ironite will give you that without causing salt to leach into your environment and cause future problems. Florida is the worst place in the world to use inorganic fertilizers for any reason.

Dethatch both of those grasses or you will get insect infestations that other areas of the country have never experienced. Bugs can take a Florida lawn in a couple of days and leave you with nothing. I like organic liquid dethatchers in Florida (microbes that eat the thatch). You want decomposing organic material on top of that sandy soil, but you don't want dead grass piling up and creating thatch. Too much thatch and you have to water all the time because your roots will be growing in that thatch and when it drys out, it stresses or even kills your grass. Even if the roots have grown beneath the thatch, that sensitive layer within the thatch will be the weak link.

St. Augustine will fill a bare area in no time during normal Zone 9 Florida weather. Toss your coffee grounds in your lawn. Anything with sugar in it. Peat moss. Rabbit food of course. Feed those microbes because they are in a thin coating on top of very sandy soil and the sun loves to bake bacteria. So keep your "grass" "fed."

Fortunately, UF extension is awesome and can do a soil test for you to nail down exactly what you need. Contact your county commission and ask them to point you to the nearest extension officer. Do ask them what Florida Friendly means. The more you can eliminate irrigation and go with natural watering, the better off Florida will be. There's only a bazillion golf courses sucking water out of the limestone aquifers, so........

Most important with the irrigation is the color of your water line. Is it purple? Then it's reclaimed water. That's good. If it's not, then you are wasting potable water. Try and harvest rainwater for use on the lawn if you can. There is no place in the world that's better for cisterned water and a solar-powered pump for lawn irrigation.

Here is a link that might be useful: Florida Friendly Yards

Thatch can be a problem in improperly cared for lawns, but some is beneficial and necessary. Grass clippings do not contribute to thatch buildup. Although I looked for information from the University of Florida the closest I could come is the link below from Texas A & M.
Thatch is greatly misunderstood or people are deliberately misinformed about it.

Here is a link that might be useful: About thatch

Someone once wrote something about leaving grass clippings on the lawn and the entire world misunderstood the correlation between that statement and thatchy grasses in Florida. lol Seriously, what works in one part of the country doesn't work in another. You gotta love academia.

Grass clippings lying on top of St. Augustine grass are not in contact with the microorganisms at the soil surface and they do not decompose easily; they sit there for weeks if left unattended, blow into the stormwater runoff and contribute to ecological havoc of freshwater all over Florida. Bag them and compost them as often as you can. Use common sense, a couple of times won't matter, but (read below) use compost tea if you leave them on top of St. Augustine or Zoysia.

As the TAMU document states in its very first sentence, "Thatch is the intermingled layer of living AND DEAD GRASS STEMS, roots, rhizomes...... etc etc etc. So believe me, the DEAD GRASS STEMS from your lawnmower do contribute to thatch in your instance. In other instances, no. But this conversation is about YOUR lawn, Florida homeowner. The water was probably set for 45 minutes 2x a week because the thatch layer in your grass was keeping the water from getting to the soil. Bad news. Dethatch until you can see your dirt. Look at the link below. It's COMMON in Florida for people to have 3-4 inches of unhealthy thatch. Healthy thatch doesn't prevent you from seeing your dirt. Learn the difference.

In Florida, you'll have plenty of dead grass stems from the natural ecosystem there on the soil level. The natural, healthy thatch layer recycles those grass stems as long as they don't get overloaded with more grass clippings from the unnatural act of cutting grass with a lawnmower. In fact, if you don't cut St. Augustine at a low enough level, you'll see grass stems 4-5 inches above the ground, and when you cut, those stems contribute to unhealthy thatch.

Adding more dead grass stems with grass clippings is not needed to provide "fertilizer" or out of sheer laziness. Worse, the grass on the surface of St. Augustine becomes a haven for sod webworms, sort of a staging area before they get into the crowns. Been there, done it. Know it from hands on experience. Saw a neighbors yard destroyed in two days. He refused to blow his clippings off the surface of St. Augustine cut high. Wouldn't collect them for the compost pile or use them to mulch his myrtles. He fertilized with inorganic nitrogen. He did everything you can do to invite sod webworm. And he got it good. What he couldn't figure out is why they didn't get into my St. Augustine. Regular, manual dethatching and core areating helps, though the areating isn't nearly as important in the sandy soil there as it is in my Dixieland clay a few hundred miles north. I do it to allow organics and soil amendments to get under the natural, healthy thatch. Ironite works great on St. Augustine, but not too much too fast (it's a waste, just dissolves and washes through the sand).

The rhizomes and stolons of summer grasses in Florida leave most home lawns loaded with thatch TO AN UNHEALTHY DEGREE. It becomes obvious. I've seen 4-inch layers of thatch in yards and a homeowner proclaiming the need to water every other day to keep the grass healthy and green. Well duh, your grass is growing in thatch, and if you don't water every other day, the roots will dry out and die overnight! Remember in the summer in your zone there, it stays above 80-degrees all day and all night. You know. That kind of heat isn't common anywhere else in the USA, maybe south Georgia below the swamp sometimes. It's nonstop for months. No cool Southwestern nights in Florida. Just heat and humidity all day all the time. St. Augustine and Zoysia love that and grow faster than anywhere else. You will replace all your plastic yard tools at least every two years since they always become brittle from the heavier ultraviolet rays we have in Zones 9 and 10.

So use organic dethatchers in liquid form. Do regular light applications. Compost tea is just awesome, so harvest rainwater, mix composted clippings in a barrel, stir often, spritz a eensy-teensy mist of malathion or vegetable oil on the surface every couple of days (prevents skeeters) and get yourself a Hozon (google it) to spread that tea around the yard. Now if you do that, you can actually leave a little of the grass clippings on top of the grass because the compost tea provides some of what the soil surface has and hence decomposes the clippings faster. You can see how fast grass on the surface decomposes as opposed to grass that's in contact with the soil where the microorganisms decompose it very quickly.

By the way, one reason you keep a circle of completely bare ground around your orange trees in Florida is because of diseases that grow in homeowner lawn thatch and the insects that need that thatch (healthy or overly thatchy) for cover from the sunshine. Insects and funguses love thatch, which once again, can go from healthy to unhealthy in Florida if you don't pay attention to your lawn and you keep your nose stuck in a book or in your internet readings. Get out there and look at the ground. If you can't see the soil, you have too much thatch in your Florida yard. And there is nothing better for composting than St. Augustine or Zoysia grass clippings. You know those live oak leaves take a century to break down, so keep them as far away from your yard as you can if you want a nice healthy St. Augustine and Zoysia grass lawn.

Here is a link that might be useful: Zone 9 FLORIDA thatch info