If I paid property tax in 2022 at closing then I paid full next cycle in 2022. Only the yearly tax is on 1098, do I add them together on my return?

It does not matter what year the property tax was "for"----it matters when you paid it.  If you paid property tax in 2022, you can enter it on your 2022 tax return.  The amount that can be deducted for property tax and state and local tax is capped at $10,000.

Yes, include the total amount paid in 2022 for property taxes. In 2023, and forward only enter the amount‌ paid during the tax year. Even if you made 2 payments in 2022, you include both because they were all made in 2022, for the 2022 federal tax year. Also, even if you prepaid next years property taxes and they accepted payment or paid early in tax year 2022, you include all payments made in 2022, even if for the 2023 property tax year.