I called the apartment complex today and got on "the list" for the downstairs apartment. I gathered that it is a short list consisting of me, but I'll have to wait until the person moves out. The guy told me that new renters only sign 12 month leases so it will be at least a year before I can move in and of course if they renew it will be longer. The good news is that even if I am in the middle of a lease he said they just transfer it to the new apartment so there will be no issue there.
So another year of dealing with the AC sound. It seems to have been bothering me more the last few weeks and especially few days, but I'll survive. I know it probably doesn't help that my TV gets used less and less and a lot of times my apartment is completely quiet except for the clacking of my fingers on my computer keyboard.
I'll try not to complain too much this year. I truly love my apartment, my complex and the area I live in. It's just this one thing that is a problem. I found this article and while it doesn't describe me exactly I feel a lot less alone with this problem than I did before (the site has an auto start video that you'll have to mute... stupid web designers).