Will I lose unemployment benefits if I quit my part-time job? Please help?

First off, I live in California. About 6 months ago I had picked up a second job (part-time, the job I'm working now) in addition to my full-time job that I had had for almost 4 years. Three months ago I was let go from my full-time job so I was able to collect unemployment. The problem is, my PT job is 3 hours a day, for minimum wage. I'm going to have to move back in with my dad who lives an hour away in about a week or so, and if I was to keep the PT job, the cost of gas to get there would eat half of each day's wages. Not worth it since my income is capped by my unemployment, meaning however much I work or don't work, my unemployment adjusts to make it the same every month. I am also actively looking for work and have started going on more interviews. I don't like the idea of constantly asking for days off, especially if it's to find another job. It seems wrong to me but I don't know what to do! I can't afford to lose my benefits because of my financial obligations like my car (which I need for work). I just finished University and my student loans kick in soon, and I won't be able to afford that in this situation! I tried calling an EDD rep at my local one-stop and he said he doesn't know for sure if my situation would allow me to maintain my benefits. My dad lives in a larger area with more job opportunities but with EDD, you never know. If you know whether or not I will lose my benefits please help!