Water. Be religious with even and consistant watering. Keep off the lawn entirely till the seedlings become actual grass. A fertilize or compost might feed the soil, but does your soil need feeding? It does not actually help the germination process, but can feed your growing plants if your soil does not have enough.

Would a light coat of soil help keep moisture? drying back is my biggest worry

Either seed free straw or light burlap can provide some coverage and moisture retention. It's what the highway folks use when they reseed after road projects. A light coat of soil will just bury your seed and dry out too.

Oh my! The fescue seed you bought should never be sold or used south of Nashville. It gets too hot for the grass to live especially in FL. Your FL turf grasses include St Augustine (only sold as sod), hybrid bermuda (only sold as sod), common bermuda (sold as seed and, rarely, sod), zoysia (seed or sod), and centipede (seed). Of those, the only one that will thrive in the shade of the trees opposite the house is St Augustine. You might find a zoysia that can live there, but I'm not familiar enough with the varieties to help with that.

If you continue to attempt to salvage your current situation, you'll have a crabgrass lawn by July. In a way, at least it's green and can be played on. Now is the time crabgrass seed is warm enough to germinate. It germinates in less than half the time fescue seed germinates and will achieve pretty good coverage by the time the fescue gets around to it. Crabgrass is a hot season, annual, grass which will thrive until the first frost. Bermuda is a hot season perennial grass which will thrive possibly all year in Jax if you don't get a hard freeze. If it freezes it will go dormant and return when the soil temps warm up next spring. Crabgrass needs new seed every year.

Adding more topsoil is almost always a mistake. If your only reason is to cover up the seed, then it would be a mistake. Grass seed germinates perfectly fine when simply pressed into the soil surface. When the pros do it they use a roller filled with water for extra weight. You can rent a lawn roller and DIY that. Or you could go out with all your friends and simply walk on the area where you seeded. The pressure from the roller is about the same as the pressure from feet walking on it. Even animals walking on it helps.

Can chopped leaves, needles, and small sticks be used to mulch new grass?

They can, but generally new grass does not need to be mulched. Mother Nature never mulches Her new grass.