How do I move pro version that I've already paid for to my new device?

You install the sniper app the same way as you do any other app on your Android device.
Just need to go to Google Play and install it from there.

And it won't charge me again? Because it sure looks like it is asking me to pay again

If you use the same Google account, then it's free to re-download. Like any other app in Google Play.
That's between you and Google. The sniper has no control over it.

Bless you my child! Is there an emoji for a deep-bow? For you guys who do this stuff everyday, it seems so simple. But for those of us who don't; we struggle. The key for me was when you said "the same Google account". I had set up a new Google account for this new device. So,back I went to do what I did not want to do. Upon reflection, this one decision would have caused me all kinds of grief down the road. So I updated my Google account information to include my other Google account, and went to the Play Store by speaking OK GOOGLE, TAKE ME TO THE PLAY STORE. I selected the Pro app, and, as though by magic, the face of the app no longer displayed $9.99. It displayed INSTALL. Hallelujah and I'm still looking for the deep-bow emoji. You not only solved my immediate problem, but undoubtedly saved me all kinds of future anguish.

You should not create a new Google account just because you have a new device.
I did not know that now Google allows you to include purchases from another account. That's very good and I'm glad it worked great for you after all.

Thank you for your follow up and good luck sniping.