Need to make 2018 Q1 estimated tax payment. We moved mid-2017. Should I file form 8822 before sending in the Q1 voucher? If so, allow how long for address to change?

If you send in the Q1 voucher with the correct address, that will change your address with the IRS. 

I would still send in the Form 8832, Change of Address Form just to be sure. 

You can always pay that estimated tax payment online to get immediately confirmation directly from the IRS. You can pay that at this link: https://www.irs.gov/payments

I'm a little suspicious of this assumption. If simply filing an ES payment voucher - or for that matter, filing a new return - will change my address, why is a change of address form ever necessary? And IRS instructions about estimated payments say that a change of address SHOULD be filed, thus my followup about the timing.
From context, it is possible this assumes that quarterly payments are in progress, and the move occurs between quarterly payments, in which case I would agree that a separate form is at least prudent. But we've never made quarterly payments, we just happen to have incurred a significant taxable event this quarter. So my concern is about doing things out of order, and having to straighten things out a year from now, even potentially owing a penalty. So what's the safest route?
The associated fees make an online payment unattractive in this case.