How do I go about starting a new federal return so I can do separate state returns?

Yes.    If you need to file a Joint federal and 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

Reading the FAQ it looks like you have to use 3 accounts.  One to file the Joint federal 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.

You will have to mail the 2 separate state returns.   

Online is only good for 1 return per account.  But you can use the same email address for 5 accounts. 

Online is only good for 1 return per account.  But you can use the same email address for 5 accounts.  You can start another return in TurboTax Online by:

1. Logging out if you are in your account
2. Choose what TurboTax Online product you want to use for the second return
3. Create your account screen
4. Set up a new login for the second return
5. Start working on the new return

The Desktop program can do unlimited returns and efile 5.  You can buy the Desktop CD/Download program directly from Turbo Tax

TurboTax® 2021-2022 CD/Download Tax Software, Compare Desktop Tax Preparation

Or buy Turbo Tax at several stores like Costco, Sam’s, etc.   Where can I buy Turbo Tax Desktop program?

https://ttlc.intuit.com/community/choosing-a-product/help/where-can-i-buy-the-turbotax-software/00/25893