What about replacing all of those 2x4's with 2x10's and either toe-screwing them in, or screw an additional 2x4 to the side of each for fastening them to the existing studs??
The problem with that is the 2x10 cupping over time, unless I got a very straight one with no heartwood. I'd probably have to back it up with a 2x4 every 2' or so.
I'll use screws and glue for the plywood on the side. That should take care of any sagging. I'll have to devise some way of securing it to the existing wall. Maybe I'll tear off the existing siding there and tie the bump out to the existing studs and plates. If that fails it will be because the wall came down.
Thank you to all who replied. You've saved me from a lot of headaches.