You unfortunately got the dreaded "pink milkshake". The transmission cooler which is within the radiator fails and mixes coolant and transmission fluid (pink milkshake). You're going to need to do a few things, replace the radiator, flush the cooling system and flush the transmission.
The tranny shop is somewhat correct that it is VERY hard to remove all of the coolant from the tranny.
Did the truck overheat? If you look in the coolant reservoir you should see the mixed fluids which looks like a pink milkshake. Depending how long you drove with the fluids mixed will determine whether the tranny and possibly the engine (if it overheated) can be saved.
Do a search on here for "pink milkshake", there will plenty to read about what people have done after it happens.
Keep in mind though, the vast majority of people who got a pink milkshake ended up having to replace the transmission even after flushing it ( I did). It could last a couple thousand miles (most likely) or 100k miles (VERY unlikely), most fail within a few thousand miles though. If the truck overheated badly, the head gasket may fail soon after also (mine did).
Sorry to be the bearer of bad news. This is a known issue with these trucks but unfortunately many people show up to these forums after it's too late.
Good luck!