Go to the first day of class, even if you're not registered, and see if the professor is willing add you. Worst case scenario is that you don't get in and you gave up an hour of your time. But you might get in, so it's worth a try.
Second suggestion, take other pre-req and general ed courses in the mean time to rack up more credits and improve your appointment date. I'm sure there are other pre-reqs besides the big three, right? And most nursing programs
encourage students to get the GE courses out of the way since there is little to no time to take them while you're in the core nursing program. If you plan on getting your BSN, you can also start taking some of those pre-reqs since there may be courses they require that are not covered for the ADN. So at least that way, you're still working towards the bigger picture.
Last option, look for a school that is less impacted. The CC I went to had very few students but the CC only 5 miles away was so overcrowded and impossible to register at. I'm in SoCal btw. I don't know why so many people wanted to go there when my school wasn't that far away. I never had to waitlist any of my pre-reqs. So if you're willing to go to a different school, and I know it's a pain, but you might have better luck.