If you have paid the first installment of taxes, due no later than December 10, but have not paid the 2nd installement due no later than April 10 and you close escrow before April 10, you should be fine not paying for the preoperty taxes separately as long as there are enough proceeds from the sale to pay the property taxes. However, even if you were to pay the taxes, it will be prorated and credited back to you at close of escrow so it will balance out either way.
All works out in the end. Escrow will prorate the property taxes by the day. Taxes start as of July 1st. The first payment due by 12/10 and are for taxes up to 11/1. So you owe them.
If you close escrow after 11/1, then the escrow company will withhold part of the proceeds from the sale to pay the daily pro rata owed from 11/1 on.