Moved to MA from CA (Nov 27, 2023) - part time resident in both states or just one?

No, you aren't a part-time resident, but you are part-year residents of both states.  You were a resident of CA until you moved to MA on 11/27/23; then you became a MA resident. 

 It appears that your income got added twice.  Delete the W-2s that you already entered in the federal section and start over. 

Follow the steps below to delete the W-2s:

1.  Open TurboTax
2.  Enter W-2 in the search bar
3.  Select Jump to W-2 link in the search results
4.  Select delete ‌next to the W-2s that you will re-enter..

For additional information, review to the TurboTax Help article How do I view and delete forms in TurboTax Online?

Next, add the first W-2  exactly the way it is.  After you add the state and local information from the first W-2, select Add another state to add the state information from boxes 15–17 from the second W-2.

If you have information in boxes 18–20 on the second W-2,  Select Add another locality and enter the information  that's in boxes 15–20.

After you complete your federal tax return and you move to the State Taxes sections, both of your part-year state returns will be listed.  Prepare the state that you moved from first, then the state that you live in.

You will allocate your income to the state that you were a resident of when you received it. You mentioned that you were in CA until the end of November, so divide your total income by 11 and allocate that amount to CA and the remaining amount to MA.

Review this TurboTax Help articles How do I file a part-year state return?  and How do I file a part-year state return? for more information regarding completing your state tax returns.  Also, ‌ review How do I allocate (split) income for a part-year state return? for further explanation.