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).