What are you trying to do?  TX has no state income tax.  The software is trying to find out where you lived and/or worked in 2020 in case you need to file a part year tax return for another state you lived in for part of the year before you moved to Texas.  And if you live in Texas now, you need to use your Texas address on your federal tax return.

There are certain 12-month residency requirements in order for a person to get to use in-state tuition at TX state universities.  But you'd have to check with your University Registrar's office to determine the specific details....especially if you are a dependent whose parents live out of TX, the residency rules for getting in-state tuition apply to the parents in that case.

Residency rules for income tax purposes may not be the same as the rules for other residency purposes (e.g. instate tuition).

In Turbotax, you put the date you physically moved to TX.

That said, what's going on? That is, what are the details of your move?  Did you move to TX for school? Do you maintain a residence back in the old state, e.g. your parent's home?  Mostly, do you intend to return to the old state?