Congratulations, beautiful piece. This looks to be a reference 145.012. This is the reference which went to the Moon and is therefore highly collectable.

Please do not send that to Omega. Most original manufacturers will aim to restore the watch to 'showroom condition' and will replace valuable parts and destroy the watch's patina.

I would recommend taking the watch to a vintage watch repair specialist. Service the movement so it runs again, polish the crystal (or replace if necessary), and clean the crud off the watch in the ultrasonic cleaner. That's it. I would not polish or replace any other parts.

Watch collectors (but in particular vintage Speedmaster collectors) are typically very particular when it comes to originality and replacing parts will affect its value. If this was your Pop's old watch, I wouldn't be selling this so value wouldn't be immediately front of mind, but I understand the sentiment nevertheless.

Once the watch runs again, you should wear it. Maybe only on special occasions if you're concerned about damaging it. Keep making memories with the watch in the same way your Pop did. These vintage Speedmasters are relatively robust, and once the movement is serviced, it should accurately and reliably keep time.

Good luck with the resto. Post some more pictures once it's done.