"I do not have a print on checks as option"

Edit Vendor Name; click on Payment Settings tab on the left. Here you see the field for Print On Checks As.

"and a new vendor means that my vendor balances are incorrect as they are the same vendor, but editing the existing vendor changes the payments I've already made"

You either pay the one name, the other name, use only one name and Fix the payee name, etc. What you start with, as I noted, is the same Entity or not?

Example:

John Doe is now running as a Corporation = new FEIN = new Vendor Name for you.

Or;

John Doe changed to Doe Cleaning, but the Name is what you use for 1099 reporting, so this is the Same Vendor and you can leave it as John Doe (because this is in Your File) but Change the payee name for their Banking to work.