What is the process to transfer a visa/extension from one employer to another in Thailand without leaving the country?

Just a point. If you are on an extension and your work permit has already been cancelled your extension is no longer valid and should have been cancelled the day your work permit was cancelled. Your old employer can't cancel your extension, you have to do that yourself.

You wouldn't apply for any new VISA at all

You'd cancel your current extension with documentation from your employer stating your termination date etc, and proof you cancelled your work permit

AND

then you'd immediately apply for a NEW yearly extension based on documentation from the new employer.

It takes you coordinating with BOTH of them because you need the documentation from the old and new employer to pull of the switch sponsors for an extension

Talk to your new employer and see what they say. Whether you can or can't do it inside the country is totally up to them and your former employer