In all my real estate transactions property taxes and home owner association dues were handled as part of the closing. The closing agent collected the information regarding amounts and due dates and then as the closing date approached let everybody know how they were to be handled.
You will only be responsible for your portion based on how many days you own the property. This part of the transaction will be handled out of the purchase and settlement funds. It is possible that you will send money to the buyer for your portion, or they will send money to you for their portion. It depends on the due date, and is the item paid in arrears.
This is part of the settlement costs that the loan officer or real estate agent can't nail down far in the future because the total will change day-by-day and they have no idea when the settlement day will be.
Because you are the current owner and many loans require funds for taxes and insurance be kept in an escrow account, that can also factor into the equation. Closing right when the taxes are due can cause an issue because the current lender will be programmed to send out the payment on a specific date, and you don't want to have to chase down the double payment.
The professionals know how to handle this and will be able to explain it to you based on your situation. They are very well versed in how property taxes and escrow accounts work in your jurisdiction.