"would I still need to amend my 2019 taxes if I placed the payment into a new 401k or IRA?"
No, as long as you roll over the entire box 1 amount.
The 20% withheld for taxes will be credited on your 2020 tax return, it's not gone. By doing the rollover and avoiding the income tax and penalty liability on your 2020 tax return, you'll get a larger refund (or lower balance due) when you file your 2020 tax return.
There are many financial institutions that offer traditional and Roth IRAs. I suggest finding one with low or no fees relative to the amount you roll over.
If you don't roll over the distribution, the point of amending your 2019 tax return is that it will not be the same because Form 8880 must change. To find out if your credit is reduced or entirely lost, prepare the amendment. Doing so will cause the credit to be recalculated; the software can't provide forms for mailing without doing the calculation. If you find that that credit remains unchanged (because the amount of your retirement contributions for 2019 were more than $2,000 greater than the amount of your recent distribution, which seems unlikely given the information you have already provided).
