Yeah you really have to be proactive in searching out for appointments the whole "healthy CA" site that's supposed to alert you to appointments (supposedly) really doesn't do much of anything I've found.
I've already had both of my shots, and granted my "eligibility group" was quite a bit smaller, but still got to be a bit of a hassle as every place wanted you to make an account, and the fun stuff was they won't tell you if an appointment is available before you start entering in all your information to make a flipping account. Not sure if rules have changed, but you probably don't need to restrict yourself to your county