Now this is what I'm talking about!

From a directory of campsites in the National Forest:

GENERAL COMMENTS:
The elevation is 7,100 ft.

The campground, pack it in, pack it out, meanders its way around
a hilltop above Joe's Valley Reservoir in a pleasant mixture of
sagebrush, Ponderosa pine, Pinyon pine and junipers, such as
Western.  Many of the camping sites have panoramic views of the
reservoir and surrounding mountains.  Most of the camping sites
are in the open, but wide spacing between them provide good
privacy.  Long parking aprons can accommodate boat trailers but
there is also a large trailer parking area at the boat ramp. 
Adjacent to the boat ramp is a caf‚/marina providing such
services as bait, ice, limited groceries and homemade bread pies.

SEASONS
Open Seasonal: Yes
Open All Year: No

Seasonal Comment:
Open May 15 through October 30.  No potable water after October 1.

RATE AND MAXIMUM STAY PERMITTED
Rate: $10 per day - single
$18 per day - double
Maximum Stay Permitted (days): 16

And what's more, Google Maps shows it and there's a Google Earth view of the area.

There are others in the Manti-LaSal area, with overnight costs in the vicinity of 7-10 dollars.  And reservations on the internet of by phone.

I know that area from my days as a whitewater guide based near Moab.  It's friggin' gorgeous.

One long day (in an RV) from Flagstaff. I think a possible itinerary is shaping up.