It's hard to find a parking place within 30 minutes of downtown L.A. for $50.... and I don't know of any state, federal or county sites that fit that description at all.
Seriously, many spots at the campgrounds are already reserved for the next 4-6 months and about all that may be left are spaces in existing mobile home parks that rent to overnighters.
I'd hit reserveamerica.com and take what you can get and ignore the prices.
BTW: When you say "within 30 minutes of downtown LA" that equates to about 10 miles between 6am and 10 am, plan accordingly.
BTW: Did you check the tire dates on your recent purchase? For that trip, if any are over 5 y.o. it,s time for new tires
before you go.... no matter how "Good" they look, and don't forget to have wheel bearings checked at the same time.