I've been using the Good Sam web site to check out various places. They have a very good rating system for the park infrastructure / bathrooms / appeal.
I don't think you'll find much space between trailers, at least I haven't, except in the state parks, but they have a 14 day limit, then you have to leave for a day. Private parks in major cities (if they're good) are full up until April. They want the long term renters, 3-months+. They'll take a reservation for November for less, if they have room.
That's been my experience thus far
We are getting ready to leave and head down there. We've managed to cobble together a month of parking in 2 different state parks. It means a lot of moving around. We have 2 5-day spots a 2-day spot and a 14-day spot. Since I didn't even start until 2 weeks ago I think I did OK. At least we'll be warmer than where we are now. We'll get to visit relatives and see some sites.
Good luck, don't give up