Hi Jen,
Since cob is a poor insulator you need pretty thick walls (like 1-2 feet) to get
enough thermal mass for a cob wall to really be effective. So you would probably need to furr out your 2x4 or 2x6 and extend the perimeter of your foundation.
Protecting the existing drywall is kind of a Catch-22. On one hand it would need to protected from the moisture of the cob during construction to prevent it getting moldy. But then you want the wall to be vapor permeable when its complete so moisture doesn't become trapped in the wall. House wrap is supposed to be
water resistant yet vapor permeable so that would probably be your best bet.
If you like the aesthetic of cob I think I would insulate the stud bays with something like wool or
rock wool batts and then do a lath and lime plaster finish on the exterior. It would look like cob but without having to make any changes to the wall system or mess with the the electrical.