something ain't right. if they're not acknowledging the $10K you would owe interest on $10,822.
your entire balance was due 4/15. any payment after that incurs interest and sometimes penalties.
pay the interest and send them proof that you paid them the $10,000. that would be the front and back of the check and the bank statement or a copy of the credit card statement payment. if you paid some other way you'll need to submit whatever proof you have. send everything with the notice to the address indicated. keep copies for your records. use a method where you can track it or get proof of delivery.
you use the term estimated payment. did you use a 2019 estimated tax voucher because the term is extension payment. and you did file form 4868.
by the way the best way to make an extension payment is through IRS direct pay. you get a receipt. there can be a problem if the routing or account number are wrong or there's not enough money for payment through bank account. credit cards the issue would be enough unused balance to make the payment
the positives are you can set up payment in advance and have the funds withdrawn on the due date. it will take 2 to 3 days for the funds to clear but if there's enough money, the IRS considers it timely. doing it this way automatically gives you an extension without filing form 4868.
Did you actually make an estimated tax payment (Form 1040ES) on 4/15/2019 or did your instead make a payment with a request for extension (Form 4868)? These two types of payments are different and reported differently on your tax return. On an 2018 tax return, estimated tax payments are reported on Schedule 5 line 66 and payments with extension are reported on Schedule 5 line 71. These are each entered differently into TurboTax to cause TurboTax to put the payment on the correct line. If it's on the wrong line, its possible that an IRS system and examiner might not recognize the credit for that a payment. Also, if you made an estimated tax payment rather than a payment with a request for extension, it's possible that the IRS credited it to 2019 instead of to 2018.
I suggest obtaining the transcripts of your tax account for 2018 and 2019 to if and how the payment was credited: https://www.irs.gov/individuals/get-transcript