Green River State Park is either in or really close to town. Its also the put-in for Labyrinth/Stillwater Canyon float so there'll be fellow boaters hanging around rigging in the mornings & evenings.
BRING YOUR BOAT!
You'll have good access to day and overnight trips from Green river. There's a day trip just below the Deso/Gray takeout floating down to Green River that doesn't require a permit. The Labyrinth/Stillwater Canyon float to Mineral Bottom (Class I but very scenic) could probably be done in four days with the flows you'll have then. If you're in the area, the drive over to Moab for the daily run there would be easy.
From Moab you could continue on over to the Dolores below Bedrock w/ or w/o a lazy, meandering float across the Paradox Valley before entering Paradox Canyon with fun Class II/III whitewater down to the confluence with the San Miguel (access), possibly continuing on through Mesa Canyon. Check on permit requirements for Mesa and downstream.
There's also the possiblity of picking up a San Juan cancellation permit from Bluff to Mexican Hat (us suave elites call it "Sombrero"
) - you'll need to have a permit app on file with the managing agency and a small party has a real good chance of getting a launch on short notice.
Have a good time!