Los Angeles here, but this California Department of Public Health website should be able to help you get a vaccination appointment scheduled.
https://myturn.ca.gov/
The first does was VERY. HARD. TO. GET. a month ago.
I got the second dose last week by scheduling an appointment via this website and I was in and out in less than 20 minutes, inclusive of the 15 minute wait after the injection.
The first dose was injected in the back of my arm / tricep area but I was also given the second dose in my forearm.
Please note that it takes time for the vaccine to take effect in your system -- see attached.