Axions can seem a bit goofy. One thing to know is that when talking about particle physics, we like to talk about particles, but "particles" are really a concept from classical physics. When doing quantum physics, one is fundamentally concerned with waves. If you've read any about quantum mechanics, you've heard of "wave/particle duality," which is something of a connection between the two pictures. Another thing to know is that axions should interact electromagnetically.

The bottom line is that because axions would necessarily be very "light" (that is, very much not-massive), it is misleading to picture them as "particles" and better to picture them as "waves." So while it's true that axions would feel the electromagnetic field of an atom's nucleus, that's really just because it's an electromagnetic field. So to make an experiment which is sensitive by modern standards, you say "forget individual atoms, I'm just going to make a cavity and crank it up to large electromagnetic field." And that's exactly what is done in practice.

To a certain floor. Of course a particle which only interacts gravitationally is still possible but that's the cosmologist's nightmare. It'd be basically impossible to prove it's existence and anything could happen (like a small but fast fairly massive ~asteroid mass clump of dark matter explaining poltergeists)