Just wanted to update the situation.
I got my TCM from the dealer this morning, and plugged it in. No errors right away (which means I'm at least that far ahead of it...). Drove it around the neighborhood for a few minutes and had no problems. Got back to the house and checked for codes, and none were there. Went ahead and mounted the TCM where it should be and then went to work. Shifts are much smoother than I ever remember, and I don't have the 3-4 delay when it's cold either.
Just for reference, the number on the module that came out of the Jeep was 5602 7564. The number in the parts sheet is 5602 7316, which seems to also be what salvage yards have. If you go to a salvage yard looking for this, you may want to tell them the 2 numbers and that they are the same. In fact, it doesn't even appear as if they make the module with the 7316 number. If you order from a dealer, you'll get part number R602 7564, which means it's a remanufactured unit. They only have the remanufactured modules now, but it seems to have (hopefully, knock on wood, say a few hail mary's and amens) fixed most, if not all of the tranny problems I was having.
Drove it a bit this afternoon, and I don't know if it's the computer(s) relearning all the settings and data, but it drove like crap. The tranny seemed to be good, but the engine was awful at idle. I needed to run to the store, and came out about 5 or 6 minutes later, and it idled like it was misfiring. Hit the gas and it was ok, but when it got back to idle, it was almost a surging feeling. When it wasn't doing that, it idled way low (just above the first mark on the tach). I switched the A/C off, and that helped. Drove it like that for a bit, and it seemed to idle fine after that.
Any ideas on the idle? The IAC is only about a year old. Could probably stand to clean it, but any other thoughts?
Hope this info helps someone with the same problems (or similar) to what I had and they don't have to deal with all the trouble I did to track the module down!