How can I check to see if my federal tax was paid and state refund?

If the fed balance due was paid you should see the proof of that in your bank or CC account statements.  The fed balance due could NOT be paid with the state refund at all ... in the FILE tab that was NOT one of the options given as a choice.

If you efiled....Was your return Accepted? First double check your efile status here
https://turbotax.intuit.com/tax-tools/efile-status-lookup/

When you efile you get back 2 emails. The first email only confirms the transmission. The second email says if the IRS (or state) Accepted or Rejected your efile.

If accepted by the state use this TurboTax support FAQ to check the state tax refund status - https://ttlc.intuit.com/questions/1899433-how-do-i-track-my-state-refund