When I run the kitchen sink, and the AC kicks on, I get a bad smell
throughout the apartment. If I run the kitchen sink, and the AC is
off, I get the same smell, just much fainter. I put a cork in the
pipe where it enters the drain pan, and run the kitchen sink and AC,
and got no smell at all. So I'm guessing it's coming from the drain
line. It's of varying strength, and I don't know if it's sewer gas or
not, but it's definitely unpleasant.

Do you have a trap in the condensate line? Should be there. Otherwise
you're pushing air out through it (traps dust which promotes clogging) or
suctioning sewer gas in (yuk).