If my state return was completed and given to me as a PDF, but I did not send it in yet, and my Federal was e-filed, can I go back to my state taxes and e-file them now?

Since the federal tax return was not e-filed, the state tax return cannot be e-filed, only printed and mailed.

Is it possible to take that state return and efile it now, after the federal has already been separately efiled?

Yes. If you already prepared the state return in TurboTax, go back to the Review/File area and choose e-file for the state return. If the return was completed outside of TurboTax you'll need to complete it again but as long as it's associated with a Federal return that was successfully e-filed through TurboTax, you should be able to e-file that state return now.