Will TurboTax update my account to reflect the receipts I attached showing that I paid both my federal and state taxes?

I'm still not completely sure what you mean by "attaching" receipts to "My Docs."   Perhaps it's just terminology.   Do you mean you stored some receipts using the My Docs feature?

The My Docs feature is a storage area as long as you are using paid Online products.    TurboTax website does not otherwise interact with whatever you choose to put there.  It's like having a shoebox under your bed, except that this is online and has more security.

FAQ:  What is My Docs?
https://ttlc.intuit.com/questions/1898861-what-is-my-docs

Be aware that unless you continue to use paid products, it's possible to eventually lose access to anything you put in My Docs.

TurboTax does not keep account of your tax account with a tax agency and does not hear anything back from the tax agencies about any refunds or payments.  So nothing is updated in that regard.    If your return had a net result of taxes due (owed), then your TurboTax account will always show that you had taxes due when you prepared the return.

If a return had a net refund after it was prepared, then an online TurboTax account will always show that you were due a Federal refund.