You could sell and redeploy you equity into a different investment as @Alex Olson advises (using a 1031 exchange to defer all of your potential tax payments), but remember that the rules for 1031 exchanges don't give you a lot of time to find a replacement asset.  

Counties normally provide an owner occupant a 'homestead exemption' which allows a reduced property tax payment.

The courts have noted that a taxpayer may establish hours of participation in a real property trade or business by any reasonable means. 

And you've got to take title as the same taxpayer that sold the old property. 

And let’s be clear, when the IRS comes knocking because a tax payer used a “diy” strategy, you will not be there to help them, free of charge of course.

Would the 1st tax payment be due on 2020 taxes (April 2021)?

The important requirement is that the "taxpayer" doing the 1031 Exchange be the same on both sides of the transaction.  

Additionally, the local taxing authority may also only provide a receipt in one taxpayer's name.If you’re each eligible to deduct the expense, you can both take a deduction for your portion of the expenses.

This might be helpful:  If a taxpayer's capacity in a given activity is more akin to that of an investor, then the activity cannot be treated as participation for purposes of deducting passive losses unless the taxpayer is involved in day-to-day management.

I think it varies by state, but in CA where I am, the tax collector's office puts a notice on your property when you miss you tax payments.