My primary Apple ID was my work one,
If you no longer work for that company, that company has already deleted your work email address. This is done as soon as you have left the company so that you can no longer log in. You no longer have any rights there.
An deleted email address, it is no longer valid, it simply no longer exists and it no longer works.
You may not be able to change that email work address because it is no longer exists.
but I also have a second email and Google account that I want to use
Is one of those two your private Apple ID?
You are sign in here . . . with your Apple ID & Password (?)
Your Apple ID is the personal account you use with all Apple products and services. It includes the email address and password you use to sign in, as well as the contact, payment, and security details that you use with Apple.
You can’t merge multiple Apple ID accounts.
What happens when you use the same Apple ID & Password that you use to sign in here?
See the following Apple articles
Sign in with your Apple ID -> Sign in with your Apple ID - Apple Support
Manage and use your Apple ID -> Manage and use your Apple ID – Apple Support (UK)