You may be able to do that if you act fast. Call the local office and ask if they will backdate your claim to the Sunday of the last week you worked full time. From that point on your hours were reduced to part time (not attempting to say the full time employer reduced the hours). Check to see if AZ has a waiting week or more - your first week or next of part time is used as the waiting period. Report the new employer and wages earned. Often the rep on the phone can backdate 2 weeks, more than that can take the approval of an examiner which can delay actual payment and any payments issued will be held pending determination. You do not mention why you lost full time - as long as it was not your fault, you should be ok.
Even if you do it using the onset of part time work, it still involves backdating the claim. The same wages will be used to establish the claim, they will still want to know why you did not apply as soon as you lost full time hours. It may still require examiner"s approval and pending payments.
The thing to consider and talk to the rep about - do you really need the $$$ from any of those weeks? If you start a claim now using the last day worked from last week, the claim is based on the same wages, but runs a few weeks longer than it would in the above scenarios. They need to use CA wages. So you can file in either CA or AZ whichever you like.