I filed my state tax and it was processed, but was the payment made at that time?

Your only proof that you paid your tax due is your own bank or credit card record.

I concur with xmasbaby0 that the only way know is to check the account you setup to make the payment from. TurboTax asks you what date you want the payment submitted to your payment source, so it helps to remember what date you set up and allow up to 3-5 days processing time from that date for it to post to your account.