We have gone for Thanksgiving week for each of the last four years. Yes, it is very busy, but that doesn't mean you can't enjoy the experience as long as you go prepared and with a good attitude. Thanksgiving day itself, if anything, is a bit less busy than the days immediately preceding it.
As for where to eat, our choice has been Storyteller's in the GCH. I like the food there. You can choose between a turkey dinner or the buffet which has a number of nice selections, some of them holiday themed. We always choose buffet. We pick that restaurant because I like the ambiance and the service has always been good for us. The whole experience just feels holidayish. Probably helps that we rarely eat there at other times of the year. It's not cheap, but it's reasonable for a special holiday meal, around $30 per person last year with cheaper kid prices. No characters.
DLH will do a special Thanksgiving meal. It is indeed very pricey, more than we want to pay, to the tune of seventy something dollars I think. I have heard people rave about this experience. It is buffet with characters in the ballroom, and the food is supposedly excellent.
If you really need characters, Goofy's Kitchen does a meal much like any other day of the year except they may add a few seasonal selections to the buffet line.
The PCH Grill at the Paradise Pier does a special Thanksgiving dinner as well. I don't believe it includes characters. I asked a few years ago when first booking Thanksgiving there, but we like Storyteller's so much I haven't asked since. PCH Grill does not normally have characters at dinnertime.
Since the princesses are coming back to dinner at Ariel's Grotto, perhaps that would be a possibility for Thanksgiving dinner this year.
There are plenty of places, both sit-down and counter service, throughout the parks and DtD that will offer special turkey day dinners. If I was to stray from Storyteller's, I might look into the Jazz Kitchen in DtD.
The important thing about Thanksgiving dining is to book as soon as you can which should be on or about September 22 this year. Sometimes they won't start the bookings until Monday morning, so it might be the 24th. The popular choices fill up quickly. Storyteller's, for instance, was mostly booked only a few days after the reservation window opened up last year.