To answer your original question, yes, you must file an income tax return in both states. It works like this:
Since your stay in NJ was intended from the start to be temporary, and in actuality was temporary, for state tax purposes you were a full-year resident of CA and a non-resident of NJ in 2020. That's because, per California law, a CA resident who leaves CA "for a temporary or transient purpose" remains a resident of CA.* That means that ALL your income for 2020 is taxable by CA.
*Page 6, https://www.ftb.ca.gov/forms/2020/2020-1031-publication.pdf
Additionally, the income you earned while working remotely from NJ is fully taxable by NJ. So you must file a non-resident NJ tax return, reporting your earnings from NJ. NJ can tax non-residents on earnings from work actually carried out in NJ.
You'll be able to take a credit on your CA return for the taxes you pay to NJ on the portion of your income taxed by both states, so in effect you won't be double-taxed.
In the My Info section of TT, enter your State of Residence as CA, and indicate that you had other state income from NJ. Don not indicate that you moved from one state to the other. Then, be sure to complete your non-resident NJ return before you do your home state CA return, so that the credit flows properly.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
