I move to the bay area for work from Colorado about two years ago so I can relate to your question well.
California has fantastic kayaking, especially in less dry years!
You can absolutely kayak year around on dam release runs. Spanky is correct you haft to drive through. As is such kayaking becomes a full weekend activity and in some ways, I enjoy it more because of that. It is great to fully escape the city to paddle on the weekends.
Our summer season is fantastic as Cherry Creek on the Tuolumne runs during this time. It is like gore with better boofs and less mank! It is a 3h drive from SF early morning no traffic and about a 4-5 hour drive home on a summer evening with traffic. Nonetheless, there is a solid bay area group that frequents this stretch in the summer.
Another dependable year around classic IV/ V run is Burnt Ranch Gorge. The drive is a bit longer 6-8 hours each way depending on traffic. However, it is perfect for a 3 day weekend.
Read about these run here:
A Wet State - A Guide to Whitewater Kayaking
From SF you can also kayak surf in the ocean (Dillion Beach) with a 30m - 1h drive.
This water year it rained in California and I did about 20 new sections of river in 8 months most withing weekend driving distance of the bay area.
It is not Hood River or Ashville but the jobs are great and the paddling is year around.
PM me if your are in the area and want to kayak.