Getting new drywall panels aligned to older ones on a wall
Do not coat the entire face of the drywall with compound to make it thicker. That is really hard to get right and will only serve to frustrate you. The way I see it, there are three options here that I would consider in your situation.


If I have to replace drywall in a room, replace all of the drywall. This is a bit of extra work, but the end result will be really good. Plus, it provides an opportunity to inspect wiring, plumbing, insulation, etc. that is covered by the drywall and repair anything else that needs it in the process. You can also make improvements such as adding outlets or switches.
Shims behind the drywall, which you already mentioned. This is probably the easiest way to get a consistent wall. I would attach the shims to the studs first. Maybe use a table saw to make some thin furring strips which you can glue horizontally across the studs before lifting the drywall into position.
Feather out the edges. The new drywall panels will be slightly inset from the originals, but at least there will not be any hard edges. As long as you are not painting with semigloss and the lighting is kind it should not be noticeable unless you already know to look for it.