Yes, most ads in Wednesday's paper will be geared toward Thursday only sales, like Meijer and KMart. A few mainly smaller retailers have requested Wednesday insertation as the ads get "lost" in the mound of ads in the Thursday edition.
From personal experience, you must go out early for the Thanksgiving Day paper. Most newspaper companies have really cut the return number back for unsold papers, making the product harder to find. Expect to pay Sunday price for the Thanksgiving Day paper. Many papers will be out on Wednesday night, so it may not hurt to check if out around 10 PM or midnight. The papers do not want to pay time and a half for pressmen and mailroom workers, and some good hearted circulation bosses want their carriers off the street by midnight to enjoy the day with family.
Most customers will receive a Thanksgiving Day edition free with a weekend subscription, I would strongly suggest subscribing at least for a weekend delivery through the holiday season, in most cases it is cheaper than a Sunday paper at a newsstand. You can cancel it in January. Remember that if a newspaper sells 100,000 copies a day and KMart gives them 80,000 ads, the home delivery subscribers are priority. Also remember that certain companies "zone" their ads to certain zip codes, if you live in the boonies and don't get a Toys R Us ad, it is likely not the papers fault, it's that Toys R Us doesn't expect you to drive from the boonies to their store. They also don't want their ads price comped at
WalMart either.