Asher, David, Jason, and Timothy are spot on with their advice and I really don't have anything to add except the tired old saying of, hang in there the best you can. Hopefully some doors will open soon and your situation will get better. The L. A. LGBT Center is a fine place that has helped thousands of people and is staffed by highly qualified people that care.
The Center only has a teen residential housing program, but as Asher stated, PATH is a GR8 place to start. I think LAHSA may take time (LA Housing Service Auth. helps w/ Sec 8 Apartment vouchers, but that takes time). Also, apply for G.R. & SNAP (food stamps) @ any DPSS ofc right away!
Call the center, you cam find the # on Google, make sure u ask to speak to mental health department they should have info, also you will need to have registered and gone through a financial screening first, so call and tell them what you just asked they should be able to help
They do have a housing resources department...best bet is to call and set an intake appt
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