It seems like different games behave a bit differently. I only have experience with Clash of Clans and Clash Royale. The basic steps would be:
1. Do a full backup on your iPad 3 (just in case).
2. Create a new full AppleID (using an email account -- it can be one that was used previously for verification purposes)
3. Go to Settings --> Game Center --> Apple ID: xxx@email.com
4. "Sign Out" and then "Sign In" with the new Apple ID
5. Create a "Game Center Profile" (with is a nickname -- possible to change it later)
6. Start the game -- it should welcome you under the new nickname. All game data should now be associated the new AppleID
If you want to make the old iPad fully independent, it's best to do a complete reset and set up as a new iPad. Even just logging out of iCloud, iTunes & App Store, Messages, FaceTime, Mail is not enough because the downloaded apps remember who downloaded them originally and whenever there is an update, the original AppleID holder has to give approval (which is annoying). So best to start from scratch and download the game on a clean device. But the game data should all be now connected with the new AppleID and most games will pull the data over (there may be a question asked if you want to do this or start the game from scratch -- careful as you have to answer the question correctly the first time!).
If you don't get the result you want, you can restore again from the backup you made in step 1.