Retired So. Cal. cop here, wasn't real big on citations unless it was flagrant or something just really stupid. It's been 6 yrs. since I've had my nose in a Veh. Code book but I think if your '37 has standard Ca. plates then you are required by VC 5204(a) to display current month/year tags. If you have "Historical Veh." or "Horseless Carriage" plates then I don't think you have to display them.
I concur with "bobjob" forget the DMV, go to your nearest CHP office and ask them, these guys are THE TRAFFIC COPS in the state and they will know.