Man, that's just utter laziness and b/s from the dealership. I'm sorry you're dealing with such half-asses that would rather screw you over for a new transmission then to just drop the pan and clean it correctly. I swear, you can't get good service anywhere these days. If I were you, after cleaning that "screen", which acts like a "filter", and getting some good clean fluid in there, and everything is fixed, I'd print this thread out and mail it to Toytota. That's just crap. Sorry: Rant over....
Before you start taking things apart, might I suggest spending $10 bucks and a few hours downloading the manual pdf's on the car from:
https://techinfo.toyota.com/
You'll find instructions / torque settings / diagrams / how-to's for every nut, bolt, clip, seal, latch, tsb's, etc.., on your truck. I pretty much rebuilt my 95 LS with the manual I pulled down from here, and it never saw a wrench that I didn't own, from the day I found out about that site. It takes a while though to get all the Pdf files, but WELL WORTH IT!
I'm not sure if your pan has an actual gasket seal between the pan and transmission, or you use some silicone based seal around the edges "pryotec" type stuff. I'd pull the manual down first to check.
1st thing you want to do though, is drain out the fluid the best you can via the drain plug. I also recommend something with a large diameter to catch the drips from when the pan comes off. Be aware that you'll most likely encounter several more ounces of fluid in the pan when you take it off. From there, you should beable to see the screen, which I'm betting is super filthy. It should be a metal screen, and should slide out. Just rinse it off, gently dry it off with a towel "not paper towel", and let it air dry from there while you poke around. If you can, try to keep track of how much fluid comes out.
From there man, I'm gonna have to defer to the manual and more seasoned guys with your model. If your transmission isn't banging around, clicking due to missed teeth on the gear, on freakin' fire, or simply won't budge, then you're best bet is to clean that screen the best you can. On a good note though, as evidenced by continuously dirty fluid, the new fluid is doing it's job and cleaning the system the best it can. But my guess is that the screen is just too clogged up for the fluid alone to clean the system. If I am right and you find a nasty screen/filter in there, I'd recommend after you've finished to do another drain n' fill a few weeks down the road, just to make sure....
I'd try and search on here if anyone has ever successfully flushed their entire system via coolant lines. Chances are you'll have several ounces of old fluid still in the torque converter area/ or similiar. On my Mazda, it drains only 3 quarts via the plug, but holds 9 in the entire system. Same for my old LS...
Get the manual! You'll be very glad you did.
For a little confidence boost, here is the link to the Mazda site where I did my first pan drop, with pictures. It was quite easy! I'm also NC211 over there as well...
http://www.mazda3forums.com/index.php?PHPSESSID=229f24336b861cf2034388dc00a05678&topic=66369.30
PS: I almost forgot a very very very important instruction: NEVER USE ANYTHING METAL TO BREAK THE PAN AWAY FROM THE TRANSMISSION!!! You'll groove the edges and will ultimately create channels for future leaks. Buy a good PLASTIC paint scrubber if it's stuck on there, and gently tap it "wedge it" between the pan and transmission. It'll break the seal and the pan should start to drop. And don't over-tighten the bolts when putting it back on. It's been my experience and I've heard from some others that it's pretty easy to strip out the threads of a transmission. Maybe it's aluminum? I don't know, but I certainly wouldn't want to find out the hard way!
and be patient, enjoy the time, and good luck amigo!