Can I file multiple people taxes with my account

No.  any return created in an account will destroy any previous one for the same year.  one taxpayer(s) return per account. you'll need to use a different userID but can use the same e-mail address for the second account.

your other option is to switch to a desktop version.  one fee to purchase no fee to e-file federal but there is a fee for state-e-filings except NY 

No, not in Online TurboTax   An online account is only good for 1 Federal return per account.   If you try to prepare someone else's, it can overwrite your own return.  Only desktop editions can prepare multiple returns.

In Online TurboTax the second return would need to be prepared in a separate account (separate User ID).   You can use the same email address for notification purposes in the underlying Profile section in up to 5 accounts, but each User ID has to be unique.

 FAQ:  How do I start another return in Online TurboTax?
https://ttlc.intuit.com/community/using-turbotax/help/how-do-i-start-another-return-in-turbotax-onli...