My husband and I were in a similar situation last year. We were both unemployed and found out we were expecting our first child. I called Covered California to see what kind of health care we qualified for. The staff there helped us get MediCAL. MediCAL is based on income, not savings. It was the biggest blessing and such a life saver.
You may be able to extend your health insurance coverage through one of your previous employers. Look into cobra: https://www.dol.gov/general/topic/health-plans/cobra
Also look into whether you can purchase health insurance through California's exchanged (ACA/Obamacare) outside of the usual open enrollment due to a "qualifying event". It looks like you can enroll if you lose your employer-sponsored health insurance.
Congratulations on your move! My husband is self employed and until recently, I did not receive health benefits through my job, so we have always purchased our own health insurance. With the Afforable Care Act, you can purchase insurance through Covered California. Usually, there is an open enrollment window to sign up, which is Nov-Jan, but when you move you can sign up for plans at the time of your move (moving states or large distances, not from one neighborhood to another). You can call them if you have questions from the website. I have found them to be very helpful. All the best!