How do I reload additional state for 2021 Turbo Tax on another computer without paying again for it?

Call customer support for help.

To call TurboTax customer support

https://ttlc.intuit.com/questions/1899263-what-is-the-turbotax-phone-number

Customer support is available from 5 a.m. to 5 p.m. Pacific time daily Monday- Friday

How to download your purchase
https://ttlc.intuit.com/community/installing/help/how-do-i-download-the-software-i-purchased-at-turbotax-com/00/26075

If you upgraded to a higher version or bought extra states they should be in your Turbo Tax account downloads, or you will need to ask support for them again,
https://shop.turbotax.intuit.com/commerce/account/secure/login.jsp?pagecontext=download

If you can't find your download, use this TurboTax account recovery website to get a list of user ID's for an email address. Run the tool against any email addresses you may have used
https://myturbotax.intuit.com/account-recovery