Preferably private room for rent (Los Angeles)

My partner and i-yes we are a gay couple - but not looking for any arrangement-are coming to LA from Chicago for a good job opportunity for me.
I'm 52, he's 38.
I was laid off in October and been on unemployment.
Money is tight.
I can only do 350 per week to start but once I start getting a few checks i can pay a lot more.
My partner is willing to help around the house too(clean, pet sit).
Hoping someone nice can help us