I did a filter and oil change on an old Cadillac and it made the transmission slip and eventually blow
Not unless you did it wrong.
If changing fluid and filter "caused problems", the trans was already damaged and would have failed anyway.
"I" would absolutely flush the trans fluid. Remove trans cooler "inlet" tube at the cooler. Let the tube spray into a bucket or drain pan. Start engine, run only long enough for fluid level to get low in pan--Stop engine the moment fluid begins to sputter out of tube. Drop and clean the nearly-empty pan and magnet, replace the filter. Install clean pan/magnet. Fill with 5 quarts of fluid, open ten more quarts so they're within reach and ready to be dumped into transmission. Start engine, dump fluid into dipstick tube as fast as it will flow. Watch fluid spraying into drain pan--when it's bright-red and virgin-clean, shut off engine, re-connect cooler tube. Start engine, top-off fluid as needed. Last time I did this--2003 Trailblazer 4L60E--I used fifteen quarts of fluid. Wouldn't hurt to have a few extra, unopened, in case you need more. You could always return the extras to the store.
Dump old fluid back into the empty bottles for disposal/recycling. The parts-store you buy the fluid from may offer recycling as a customer convenience.
Dextron III transmission fluid for the transmission. Rear end and front end are 80w 90 gear oil. If you use synthetic gear oil then you do not need the limited slip additive. Service the transfer case while you are there. It also takes the dextron III transmission fluid.
There's no "t" in Dexron.
There's no need for limited slip additive no matter what differential lube he uses. In fact, if his Service Parts ID tag on the glovebox shows "G80", he should
avoid limited-slip additive. In this case, I'd drain the front axle before refilling--might as well get all the old lube out of the front, along with any wear particles that have accumulated.
Agreed on drain 'n' refill transfer case.
I'd also flush the power steering fluid, and maybe add a PS filter in-line with the return hose.
'Bout all that leaves is engine oil and coolant; along with air and fuel filters.