They won't grow well if there are too much shade. If possible, prune lower branches to allow in more lights or take out tree(s) that you do not really need for anything.

How high are you mowing them? They do best if mowed high like 3-4 inches.

I am mowing about 4" high.

I've got the same problem and, at one time, did a little research on the more shade tolerant varieties of St. Augustine. I don't now recall the specifics, but it seems like these types are more prone to fungus and other diseases.

I know what the solution is, and have bought a 14 ft. step ladder and a gasoline powered pole saw. I just can't seem to work up the energy for it.

St. Augustine is actually a full sun-loving grass that just happens to tolerate shade better than bermudagrass but still does its best in full sun. It's a tossup between having thin or no grass or removing lots of limbs from your trees. I'd just plant something more shade tolerant than the St. Augustine. In Texas, everyone thinks it has to be bermuda or St. Augustine and don't usually consider outside the box. Turffalo is supposed to be shade tolerant and grow in 80% shade but I have no experience with it. It is sod only and if you can afford it, it sounds like the way to go.

Here is a link that might be useful: Turffalo