The Colorado above Moab has some unimproved BLM camp sites on river right, if you can get 'em.
Deso in three days is pretty ambitious, although if the Neferitti Road is okay, you could make Nefertitti to the ramp last three (very) short days. Deso requires a permit, but the Nefertitti run doesn't.
Green River to Mineral Bottom is a very pretty float, but just
a float, and the shuttle is long.
Dolores Slick Rock/Bedrock is a perfect three day trip for family, but no water this year.
There are some reaches on the White that might also suit you, if you can get the flow just right. USFS has info in it, I believe.