Hi to ScampingCats and welcome to the forum, 

There are over a dozen blm campgrounds near Moab, depending on how far you want to drive. The first one that comes to mind is Goose Island about 10 minutes from Moab and right on the Colorado river. It is first come first served and open year round, but depending on the size of your party, there are reservable group sites.

Also in the area within easy driving distance to Moab is willow flat cg in Canyonlands np and Dead Horse Point state park. I believe that they are open year round also. Devil's Garden cg in Arches np is awesome but not open all year. There are many private cgs in Moab but I have never used any of them so cannot comment on them.

Rather than have me tell you about all of the different options, if you have a smartphone I would recommend the Park Advisor app, If not, you can log onto their website and get the info. There are several different apps that can be used, but this one is free and works reasonably well.

Enjoy the Moab area. There are many things to do there, There is a good reason why it is such a popular area.

Wayne