Brian has a point. Legally the previous company owns your email. They pay for
the licensing required in order for you to do business on their behalf.
Unless you have express permission to keep the email, you can't. Business
email is not your private property, it belongs to the company you work for.
This could also be a conflict of interest.

However, that wasn't your original question and is off topic.

You need to point your new email account to the new Exchange server and add
your PST as a personal folder. However, if this is email that belongs to
another company, you need to ask permission. If you are merely moving around
within your original organisation this is a moot point. The Exchange server
should be the same unless you have transferred divisions, locations, etc.

Your current Exchange Admin or Systems Admin would be your best resource to
assist you with the best practice for performing this task.