Cottonwood Canyon Road is great, but stay away from any of these unpaved roads when wet (including when snow is melting during the day) - otherwise you might return without your car, even when itīs a 4WD.
Yes. I turned back from reaching the trailhead for Cobra Arch at a wash in Long Canyon Road which had been trampled by cattle.
Accessible tho were:
1. Paria Movie Set - great views, not spoiled at all, lovely hiking nearby to the west, but don't get lost in the arroyos.
2. Wire Pass - narrow, atmospheric, but easy slot canyon. Also accesses Buckskin Gulch.
3. Lick Wash - down the Skutumpah road, which is snow-plowed that far up across private land, but not all the way thru to Willis Creek and Cannonville. Note nice glimpses of Bryce Canyon cliffs to the north as you drive along.
Btw I'm happy to plug this house in Kanab which I rented: http://www.vrbo.com/365794
Also note there's a late-opening liquor store a few minutes south of Kanab just across the UT/AZ border. Reminds me of driving to Anglesey to get a drink in N.Wales on a Sunday in the 1970's.