boflex,
Refer to the links below. One addresses availability for work and school, and the other has information regarding part-time employment and collecting UI, and may also answer some other questions you may have. You can also search the CA EDD website for more information.
Able and Available AA 40 - Attendance At School Or Training
http://www.edd.ca.gov/unemployment/pdf/UI_FAQ_Tips.pdf
I say go ahead and apply for unemployment benefits, since you work part-time (assuming that you had a previous full-time job that ended, and your benefits will be based on those earnings). The worst that can happen is that you don't qualify for benefits, at least at this time. If you end up getting let go from the part-time job you currently have, apply at that time.
With respect to school, make sure that you explain to them that you are willing to quit school for a full-time job, or that you can work full-time at night, or that you are willing to drop down to part-time school, or take classes at night, or whatever you are willing to do to make sure you can not only look for full-time work, but can accept full-time work. Your availability for full-time work is the key thing they are looking for.
Good luck!