I have similar meat issues, and I haven't run into problems at WDW. Also, the portions are normally large enough that I can cut off any fatty parts, if there are any, (and the surrounding meat) that make me cringe. The only major difference is I don't do burgers, at all. Just the smell of ground beef makes me gag. Though I do enjoy the occasional hot dog, but only all beef the other kind are nasty...so I don't normally get them while out, except at 5 Guys, because they're Kosher, I LOVE Kosher hot dogs.
I definitely suggest looking at the menus. If you're really worried, then order things without meat in them.
The biggest issue I had was at 1900 Park Fare (dinner). In addition to the cuts of meat not being very good, they weren't kept warm very well. It wasn't a big deal to pick some other items from the buffet though.
Biergarten also posed similar problems, but I got my fill by eating bread and desserts.
Good
Honey Chicken at the CS place outside Yak & Yeti
Lemon Lavender Chicken at CRT
Filet Mignon at Le Cellier (my favorite Disney meal ever)
Pot Roast - 50s Prime Time - soo tasty, though I was a bit afraid to not clean my plate
OK
Steak at Coral Reef - it was a little dry, but the cut of meat wasn't bad.
Chicken Sandwich at Sci Fi - again dry
Doable, but probably not again
Chicken at Jikos. It's on the bone, but there's so much meat that I didn't have to get down to it and it wasn't as greasy as a lot of meat still on the bone. My main issue with it was how spicy it was...
Chicken Pot Pie - Rose & Crown - I can't find it on the menu anymore, so now I'm thinking I ordered the pot roast. Either way I was not impressed by the meat
That's all I can think of...ultimately I LOVE Disney fries, so no matter what I order as long as it comes with fries I'm happy.
I'm sorry very few people are taking you seriously. I have a suspicion that I'm a "super taster" (it's a real thing, look it up), which means every taste is far more powerful. I can literally taste fat in my food, so if I see any visibly I have a hard time convincing myself to take even a bite. If something is on the bone it seems to have a higher fat content, or at least be very greasy, which is basically fat in liquid form, so I avoid that as well.
Then there's a whole texture issue, so if food isn't cooked properly it becomes difficult to chew, which keeps the taste in my mouth longer.
I didn't even eat steak until a few years ago.