How to convert free version of Resolve to the paid version

I downloaded and installed the full version of Resolve on my "work" computer, and also downloaded the free version of Resolve on my personal MacBook Pro. This was in mid-2022.

I want to replace my "free" version now on my laptop with a second "full" version.

Question #1 - before I install the full version on my laptop, should I remove the free version?

Question #2 - how can I recover my "product key" for the paid version?

Yes, you should remove the free version first.
For the paid version you should either have a card with the Activation code on it, or should have received an email with it on. There is no other way to recover the code. The card should be looked on as cash. As an example, here's mine with the important sentence highlighted.

Activation Code.jpg (237.53 KiB) Viewed 1301 times

If I misunderstand you and you are asking about de-authorizing one computer so that you can authorize the second computer, you don't have to as you can have two computers activated at any one time.