Can you hold an ID card from one state and a driver's license from another?

Let's say I am in college in California, but I legally remain a Delaware resident. When I try to go into a bar or club in California, the bouncer makes a fuss because he doesn't know what a Delaware driver's license looks like since he hardly sees them, and he umes it's fake. Usually they let me in, but sometimes they ask me more questions to verify my age. I would like to know if I could avoid the hle by just getting a California ID card from the DMV to use over here, without giving up my Delaware license.