Quick and easy (also free) test: put your spare on in place of the worn tire. If the pulling and handling issues disappear, then you know your front ends is okay.

I've personally experienced some of the most weird handling and pulling and jerking/shaking issues caused by a bad tire. When the belts separate in the carcass (often not visible to the untrained eye) all sorts of problems can result.

Good luck, I hope it's just a shot tire.