You might not owe capital gains on the sale of your primary residence. The details are here.
https://www.irs.gov/taxtopics/tc701
If you don’t get a form 1099-S AND you calculate that you would have no capital gain, you don’t have to report the sale.
You don't pay capital gains. You receive capital gains. Capital gain is the profit that you make when you sell an asset. The capital gain is income. You pay income tax on the capital gain.
You pay tax on the gain from selling your primary residence in the year that you sell it. What you do with the money, what other homes you own, or where you move to, have no effect on when you pay the tax or how much tax you pay. As Bsch4477 pointed out, if you meet all the requirements you might qualify to exclude from your taxable income part or all of the gain from selling your primary residence. The requirements for the exclusion do not include what other homes you own, where you move to, or what you do with the money from the sale. If you have to report the sale (even if you qualify for the exclusion of gain), you report it on your tax return for the year that you sold the residence.
If you have a large taxable gain on the sale, you should make an estimated tax payment instead of waiting to pay the tax until you file your tax return.
What you might be thinking of is that up until 1997 there was a tax provision that you could postpone (but not eliminate) the gain from selling your primary residence if you used the money from the sale to buy a new primary residence within a certain period of time. That provision was removed from the tax law 26 years ago, in 1997. It was replaced by the exclusion of gain that Bsch4477 referred to. Since 1997 the exclusion of gain is the only tax break for selling your primary residence. As I have said, the exclusion of gain has nothing to do with what you do with the money from the sale, where you move to, or what other homes you own.