The only reason why the advance lever on an AE-1 won’t move is because the advance lever had already been advanced, which means that the shutter has been reset and is ready to fire. You have to press the shutter button to trip the shutter, then you can advance the advance lever. If pressing the shutter button doesn’t trip the shutter, then either your battery is dead or the electromagnetic shutter system is faulty.
Buying a “new” battery, even one that isn’t expired does not necessarily mean that the battery is actually good. The battery in the AE-1 is not used in many different products. As a result these batteries will sit on the shelf for a very long time. Buying a dead battery like this is a very common thing to happen, especially when people buy their batteries at drug stores like CVS. You must check the battery at the store to ensure that it’s still good before you buy it. Personally, I always recommend that people buy these batteries at real camera shops because they sell a lot of them and you’re almost guaranteed to get a fresh one. Use a volt meter or take the battery to a real camera shop and have it checked.
If the battery is good, then the electromagnetic shutter system is faulty. Canon uses a magnetic system where the shutter is held by a magnet. Only when a current is applied is the battery’s polarity reversed thus releasing the shutter. Well, these systems will get dirty over time and fail; after all the camera is nearly 40 years old.