married, living and working in different states; want to file jointly federal; 1 child, day care, tuition interest

I don't know.  Some states follow federal status.  Here's all I know  about filing Joint federal and separate state.  Is this for 2021 or 2022?   2021 efiling and online versions are closed.

See this article on how to file a joint federal return and file separate state returns…..
https://ttlc.intuit.com/community/using-turbotax/help/how-do-i-prepare-a-joint-federal-return-and-separate-state-returns/00/25564

You need to do 3 separate federal returns. One for Joint and 1 for each spouse to do the mock separate federal return to do the separate state returns.

If you are using the online version it will be cheaper and easier to do it in the desktop version. Contact Customer Service to give you the desktop program download.

Reading the FAQ it looks like you have to use 3 accounts. One to file the Joint return. Then you each need another account to fill out fake federal MFS returns to do your separate state returns. So you will have to PAY for 3 online accounts (3 federal returns and 2 state returns). That's why it's cheaper to use the Desktop installed program. If you don't have a computer to install the desktop program on maybe it would be cheaper and easier to go to a local accountant or tax place.

Q. Can they use CT address on federal return?

A.  Yes.  You can use either residence as the address for the joint federal return.

I cannot answer specifically, for your situation. Most states allow you to file a joint (MFJ) state return even when only one spouse lives there and/or works there.  It's just a matter of allocating the income, to the proper state.  Some states (e.g. Ohio) even mandate that you file the state return as MFJ if you used MFJ on the federal return. 

Thank you