It's a tough call this year with the low snowpack. Without rain to help it's hard to say what will be in. Keep your fingers crossed for rain or snow before then.
You might also want to look at both forks of the Cal Salmon if flows are high enough, as well as the Scott River and Clear Creek into the Klamath, a true classic up there. Wooley Creek as well if you don't mind hiking.
I'd also look for Box Canyon of the Sacramento River, a very nice class IV run with easy logistics.
Bring overnight gear, hope to get lucky and put the Illinois on your radar as well as Mill and Deer creek just a touch (1hr) further south of Redding.
If things really dry up, put the Upper Klamath on your radar too. As long as the roads are not too wet it makes a nice bike shuttle and has flow every day.
If all have water this is what my priorities would be in order of what I think is classic/memorable:
Clear Creek into the Klamath
Mill Creek (in a large part for the scenery, if possible read the book Ishi first)
Scott River (if flows around 2,000)
Prijon USA (coming back!)