...there's no lodging between Rawlins and Lander, if the reports of the Jeffrey City motel closing are true, and it's about 120 miles. If you hit a bad headwind, like we did, you're going to be hurting. I'd recommend you plan on two nights either side of that stretch to rest up before and after that ride.
I didn't read the entire thread, so if this has already been mentioned then please disregard. I rode the opposite direction (west to east), and stealth camped every night, but if you don't mind 'roughing it' for a night (it was a luxury for me) you can stay for free in the Sweetwater Station rest stop. There's drinking water, restrooms, picnic tables, etc. However this obviously isn't an option if you don't have a sleeping bag and aren't interested in camping, but I think it would be a shame to ride the TransAm without camping at least a couple times. The places I camped are some of the most memorable parts of my trip!
However you end up doing it I hope you have an amazing time and I wish you the best of luck!