Can my iPad and iPhone share iTunes, but have different game centers? Trying to merge I have an iPad tied to my work email for everything. I have an iPhone tied to my personal email. My work email is about to go away, but I'm keeping the iPad. I have a number of games in use on the iPad and don't want to, for example, reset Candy Crush from level 2057 back to 1. 

Is it possible to keep the iPad Game Center, even if I switch the overall device to use the personal email? Is there a password tied to Game Center that I better reset so I remember it before I change the device email? I don't think the iPhone has any worthwhile Game Center content, so if I have to abandon it I don't care.

Can you give me a precise sequence of steps to follow to ensure that the Game Center on the iPad survives, while changing all other identities on the iPad to the iPhone ID? Note that I don't think either device has any paid content in iTunes if that simplifies anything (iPad maybe one app, but nothing expensive).
Yes! Just go to Settings - GameCenter and you can sign in using the applicable Apple ID you want to use for that service.

You can use a different Apple ID for each individual Apple service (iTunes/App Store, iCloud, iMessage, FaceTime, GameCenter)–you just need to go to their settings page and sign in with the one you want.

Do keep in mind that some games use iCloud to store their game data, which may affect your plan.