I'll start with this: There is a 98% likelihood that there will be fire restrictions by then. That means that you will not be allowed to have a camp fire regardless of where you set up camp.
Next bad news: Technically, a dog must be leashed at all times while on National Forest Land.
And finally, a friendly reminder that shooting is not legal in Arizona within 1/4 mile of any forest road, creek, structure, or other campers/vehicles.
Now onto the info. There are several creeks in the Payson area, though if it's accessible by truck, there will be people, and usually plenty of them. A good start is to take the Control Road, which runs below and parallel to the Mogollon Rim. It crosses several creeks including East Verde, Webber, Bonita, Tonto, and Horton. Again however, the creeks will draw people, so loose dogs and shooting are not usually a great mix. That said, I'd recommend easy to get to spots along Webber Creek by the Boy Scout Camp, and along the East Verde in the Washington Park area. But be there early on a Friday or finding a spot may be difficult.
Another alternative is Haigler Creek (official camps at Haigler Canyon and Alderwood) or Canyon Creek (Canyon Creek or Airplane Flat) east of Payson on hwy 260. Again however, the water draws people, so solitude is unlikely. Though these two creeks at least offer the opportunity to get away from camp and be totally alone, whereas the Control Road area is much more "congested" relatively speaking.
The final alternative is to ascend to the top of the Mogollon Rim and drive along the Rim Road (FR300). Pick a side road and head north. The further you drive the more secluded it will be. There are several lakes that are stocked with fish and are popular. It takes about 45 minutes to get to FR 300 from Payson on either side, and you can easily drive up to 90 minutes on the roads up there getting further and further away from it all.
Below the rim, see Tonto National Forest's
website for info on the general areas by reading about the official campgrounds. You can usually dispersed camp if you are more than 1/2 mile away from an official campground but each one will have specific notice of restrictions.
Above the rim, check with
Coconino NF (west side) and
Apache-Sitgreaves NF (east side) for details on the lakes up there.
Have fun. Be safe!