We stayed at this place (it's a few minutes outside of town:

http://www.archviewresort.com/tentrv.html

It's less than $20 a night for a tent site. It's not lavish, to say the least, but it's cheap, and puts you close to the road for Canyonlands and to the entrance for Arches. It is a 10+ minute drive to get into Moab though, if you need grocercies, grub, etc.

I wouldn't stay there alone though in your case, just to be safe. It's far enough outside of town that there could be an elevated risk (granted Moab is a pretty tame place).