I thought the crowds were fine the Saturday and Sunday after Thanksgiving 2011...not super-light, but not at all what I'd consider "crazy." This was pre-Cars Land so crowds may be a bit bigger this year (certainly Cars Land will be plenty busy).
By Sunday night, crowds were very nice. From the timestamp, I think this was about 8:12 PM, but there was a time zone difference and my camera is a bit off on the timestamps. Fireworks are at 8:45 PM per my times guide and it shows Roger Rabbit as available, but I thought Toontown closed early for fireworks? Anyway, I don't think this is any LATER than 8:12 or so.
I got this FP for Space Mountain at 9:50 PM. I've gone in summer when all the FPs are gone by like 2 PM, so this amazed and delighted me. It's POSSIBLE SM had a breakdown earlier in the day that kept people from getting FPs, but to my recollection the FP line was not longer than usual that night, and you can tell the standby line was fine. So I don't think that was an issue.
I have a really crummy, near-unreadable, wait board picture from about 1 PM Saturday at
Disneyland. It looks a lot like the ones I'm seeing for this summer at Disneyland, which is certainly not empty, but is a bit lighter than most folks expected.
Again, with Cars Land in the picture, this year COULD be different, but I'm very happy with the choice we made in 2011.
By practically every account I've heard, the days BEFORE Thanksgiving are a madhouse, so I would avoid those if you have an option. (Though if your options are 5 days, with a couple on or before Thanksgiving, or JUST the Friday-Sunday after Thanksgiving, I'd definitely go more days.

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Have a great trip!