I have an update! I've got my set-up (mostly) all figured out now!
In addition to getting some new hides and replacing the substrate, I found out that with the UTH and the heat lamp it was getting WAY too hot in there. This explains why he spent all of his time in the log that was in there, because it probably had the best insulation and protection from the heat.
I finally bought a thermostat to control the heat lamp, but I have the UTH plugged into the wall normally. So the lamps turn on and off to control the radiant heat but he has constant belly heat when he's on the tile, but he can also get away from it by going into his moist hide. Because the lamps turn on and off, I've started being better about opening the blinds in my apartment during the day so that it can help him regulate his sleeping patterns (with it always being dark and no lamp I was afraid he would think it was constantly night). His enclosure is on the opposite side of my room as the window and I'm on the shady side of the building, so I'm not worried about him having too much direct heat or sunlight, it just makes the room brighter.
I also put a thermometer on the cool side of the tank, and it stays a couple of degrees warmer than our apartment. Since it's summer and we don't have AC, the cool side is right around 77 F. That may be too warm but with no AC I can't really control it. From September to May it should be around 71-71 F pretty regularly.
I don't have a humidistat, but I do spray down the sphagnum moss in his warm moist hide every evening while he's out and about, and I change the moss every 2 weeks or so. I've also been feeding him ****roaches periodically and sometimes some crickets (but I really hate having to rip the legs off for him so I don't think I can make that a habit) in addition to his regular meal worms.
He's eating WAY more (6-7 giant meal worms EOD instead of 2-3 in a week), is much more active, and is more comfortable with me when I wipe down the tile and spray his hide. He won't climb onto my hand by himself, but he doesn't wiggle and try to get away if I lift him up (I haven't taken him out of the tank except when I sanitize and deep clean it; I just lift him out of his hide and set him on the tile if I need to clean). If I put my hand down for him to climb on he does lick it, and he doesn't run and hide like he used to. All in all, he seems MUCH happier (at least for a gecko, haha).