Does anyone have advice on how to work the airline systems and get a good rate on an airline ticket (domestic) over the Thanksgiving holiday?

Thanksgiving is one of the busiest, if not the busiest, holiday travel periods in the airline industry. It's so busy in fact that every TV newscast on the Wednesday before has the ubiquitous story showing one of their reporters out at the airport standing in front of massive crowds. That should tell you the airlines know this and rarly offer bargain prices for travel over the holiday. All that said, you might be able to get a modest "break" on fares if you are willing to travel a few days (say Monday or Tuesday) before Thanksgiving, Thanksgiving Day or that Friday. Your best bet is to keep checking the various airline and travel sites for fares to your planned destination on a regular basis and when you find a fare that looks good to you book your flight immediately. Low fares have a way of disappearing quickly.

Depends where you're going as well. I've purchased cheap tickets to Europe over the past several years taking an extended Thanksgiving weekend.

Fly on Thursday and come back on Saturday or Friday.

Fly on Thursday, return on Tuesday.
Your competition is college students, most of whom leave early Thanksgiving week and try to return Sunday or Monday.

Be aware that some airlines seem not to give advance seat assignments on weekends like this except to their frequent flyers. Some people book multiple flights "just to be sure", so there is a lot of overbooking. So if you are one of those people who "has" to know their seat assignment in advance, it could be a problem for you.

I would be astonished if you found a bargain Wednesday or Sunday, but stranger things have happened.