Aside from the new tires and the alignment that was done when they were installed, have you had any other work done on parts of the suspension? What you're describing indicates a camber issue, probably with the front tires. Regardless of what the shop says, I'd take your Avalon to another shop for a second opinion. Case in point:
A few years ago I had a new set of tires installed on a car and an alignment was performed at the time. I knew as soon as I drove away there was a toe issue. Took the car back, and the shop checked it out swearing the alignment was fine. I insisted they throw it back on the alignment rack and redo the job. Drove away after that work, same issue. Drove straight to another shop where they discovered that both front wheels were toed 4 degrees to the left.

Moral of the story is, just like your own health where you get a second opinion from another doctor, do the same for your car because not every mechanic or shop is either competent to handle the job or experienced enough to properly diagnose a problem.