Any green dot road it's OK to ride on. No green dot, no riding.
This includes the Naneum State Forest which is mostly DNR land, with WDFW land sprinkled in, also includes the WDFW lands out to the east....the Colockum and Quilomene wildlife areas.
It is fairly actively patrolled by both DNR and WDFW. They will pop you for no helmets on quads, your license stickers, campfires if closed to fires, etc....
The DNR has a fairly good green dot map for the Nanuem State Forest on their website.
I would suggest camping up Coleman to the west of Cooke. There are a lot more opportunities to camp in and around Coleman than Cooke with quite a few creekside camps in Coleman.