The Shoeshine this time of year is still running around 1,000 cfs. This means some nice class IV on the top sections and a 13 mile run of class II below. There are no scheduled "releases", rather it is one long release from April to October, with 1K being the nominal summer flow this time of year. Depending on how much later you come up, it might drop, and there is no way to know when that may be, but it usually happens when different crop harvests start, (barley now, beets in the end of Sept).
Camping is okay, there is some BLM dry camping on Cedar Mountain On the west end of town or some "Pay" campgrounds in town. National Forset is about 23 miles west of town and has great spots but might be too long a drive. Othe sites exist but would be easier to explain in person.
I work at Gradient Sports in town, (boat shop) and can help you connect with folks. You can get on Facebook and go to the Cody Kayaking message board to hook up with folks. We usually boat after work during the week and mid-day on weekends,
You can get a hold of the shop here: Gradient Mountain Sports, paddle sports, Cody, WY