Update. My wife was getting worried so we went to DMV yesterday. They were very helpful and were able to get her CA tags but also maintain the registration and title in CA. So she now has 2 sets of plates and DMV says to switch them when she switches states. They tried to take the title/tag several times out of habit so pay attention. Ultimately they wanted the FL registration which she gave to them and then ordered a new one on the FL DMV website.
It cost $211 but the ticket for out of state tags is $300 and you can get more than one! She was not required to switch her drivers license or insurance from FL.
I called CA DMV a few weeks ago and they insisted we would need to transfer/surrender the title and tag. Internet searches did not mention that you can legally have tags in CA and FL at the same time. Hopefully this info will help someone else.