I've spent a lot of time in that country. This time of year, there will be a lot of water. For myself, I would probably hike primarily (perhaps exclusively) in water sandals (Chacos), which are great for the water, the sandy paths and the sandstone in that region.
You might focus on the Escalante drainage. For example, if the Escalante is your trip "artery", you can hike up drainages, cross the benches, drop into the next drainage and hike back to the Escalante. Maime Creek, Deer Creek, Boulder Creek, Harris Wash, Coyote Gulch, and others will all have water this time of year, and the canyons are spectacular. It's hard to go too far wrong. Once you get away from the bridge at highway 12, you can find some lonely, beautiful spots, freshwater springs, etc.
ULA has Brian Frankle's trip report of hiking this country. He blew past the Escalante drainage, but if you're looking to explore canyons you can meander there for a long time without duplication.
You might call Escalante Outfitters located in Boulder UT. They would have a sense of water levels--some of the drainages may be difficult to hike early in the season. Also, flow rates are recorded on the internet, so you could probably check yourself, though without having a basis of comparison, the flow rate won't tell you much.