Did the check clear? If paid electronically, were the payments debited, did you get a payment confirmation?
Other than those methods, IRS just considers it received as of the date mailed or payment processed. If it isn't received, they'd open an assessment after processing your return and then you'd submit the evidence that you paid the tax in order to cancel it.
Is this inquiry perhaps related to the partial government shutdown and the January 15th estimated tax due date for 4th quarter? I would not be surprised if IRS is delayed in depositing these checks! Hopefully, whoever is still there in the mail room is at least segregating incoming mail by date received, for later reference when they re-open and it eventually get processed. If not, keep any evidence you may have for date of mailing.
I recommend that people establish an account with the IRS. Then they can just log in and see their payments, and they can easily get tax transcripts. You can start the process to get an account at this link: https://www.irs.gov/payments/view-your-tax-account