i dont know you, and i dont know how this website works at all,
i just know 2 things,
1- i'll try to give advise once in a while if its ok with u guys
2- i know atv's
i happend to come across this site when looking for a free download service manual, so far i havent find any, well, thats the web for ya....
Back to trx no wheels spining
you lack some info. Like is it a trx 300, trx350 etc.....my best bet is a trx300, cadillacs of the dirt roads
you mention no engine misbehaving, and never mention any kind of noise, thats goood
loosing an universal shaft or the differential whould make significant noise, and notice weeks ago, unless you were listening zztop on your walkman (my favorite)
so this is my best bet. Hope it is
Go to the back of the atv. Get hold of the rear rack and crouch.
Now move the bike left and right...
See any movement on the wheels compare to the axle or the rest of the bike???
Now you may notice some clicking noise
if so, you need a new axle and hubs, hub nuts where a bit loose, dirt and friction did the rest
if you dont have the resourses for new parts, you can go to a welding shop, your friend next door may have a mig or an arc welder, take wheels off, take hubs off, clean the splines, put hubs back, tigth the hub nut as much as you can with impact whrench, now weld hub to axle around the inside,(brake side and differential side)
now weld 4 good poins around hob to nut side, just to keep it balanced, if not, the weld inside will brake
this weldin is easy to take off later, by means of hacksaw or any power tool, and will be able to install new parts (cutting just the welding, not the axle right?)
hope this help you
let me know if it did