I am going to play devil's advocate. As I read your story, I wonder, did you really move to NJ? You packed up all your stuff, changed your license, car insurance, voting, got a place for a month with intention to stay and be a NJ resident? It really does not sound like you moved to NJ. It sounds like you went home for a visit due to family emergency and stayed there. Then you went home. Let's look at both sides of this coin.
If you really did move to NJ, then you are a part-year resident return and must file all income earned while living in NJ.
However, it you were just there visiting, all your stuff stayed in CA, your home and stuff, license, voting and intent to live, all CA, then you were not a resident of NJ. Non-residents of NJ only file tax on income earned from a NJ source. You had CA income, there would be no need to file a return.
So, you need to decide if you were actually a NJ resident or just staying there dealing with family. Part of being a resident is intention. It was never your intention to make NJ your permanent home from the way it sounds.
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.
