I've seen this problem before but on different cameras. I'm a fairly "new" photographer but I know my way around my little F3. I was working on some shots one day and all of a sudden the film advance lever just stuck. I tried advancing the film myself but that doesn't help. There is also a problem with the film counter and I don't know if the two happened at the same time or if it's just a coincidence or what. It was my fathers before I had it so I'm thinking the counter is just a side-effect of it's age but if somebody could help me with this, I'd really appreciate it. I'd rather find a way to do it myself than to spend tons of money in a shop.
Have you tried the mechanical release on the F3? Its the small lever w/ the memory button in the center. It is next to the mirror/mount area. If the shutter releases, it may be weak batteries.