Well that would help, but I was assuming stripping the wall back to the studs. An overlay sounds easier, but would be fiddly to deal with electrical sockets and the door frame, plus you'd only have 2" insulation, not the full thickness of the wall.

Honestly, it wouldn't be a big job and I could do it myself, with a hand on the hanging of the new sheetrock, in just a couple of days. I can't see a decent handyman taking more than two days. A fast worker and his "mate" could do all but the sanding of the joints in just one day.

BTW For you, and any other readers, I stripped and insulated a kitchen wall in our house in London, and the ambience of the dining room beyond was changed markedly as the insulation didn't only reduce transmitted noise, it also deadened the wall and reduced reflected (echo) noise very noticably. In fairness I had had to strip the plasterboard off the kitchen side anyway, as part of the kitchen remodel

Oh ok, I was confused by the extra 2" (didn't know what to do with it ). So it's just widening the gap between the walls so you can stuff more insulation in?

I did the sheetrocking in our bathroom back in the UK. Our bedroom wall here is approximately twice as high and four times as long though! Also tiled over so I didn't need to plaster it. It would be a good opportunity to put sockets where they were actually useful, instead of behind the nightstands where they push said nightstands away from the wall if anything is plugged in.

Speaking of which, has anybody tried those 110v sockets with USB sockets built in? I've seen very mixed reviews.

Yes, though it's more beneficial for thermal insulation in external walls than sound insulation in internal walls. 3.5" of insulation for sound insulation purposes is likely sufficient.
If you're doing it yourself, or have a cooperative handyman you could put them higher up the wall where you could reach them, near the nightstands. We have a heavy bed headboard with a shelf on top, and I wish we had sockets up there for the lights that sit on the shelf.
I was thinking of getting a couple, but hadn't realized they had mixed reviews.

Mixed in what manner? They look simple enough.

Some like them, some say they fail after a few months - perhaps the good reviews were posted before failure. Some low power so they don't charge fast enough. However, looking again, there are a lot more positive reviews than I thought so either I was mistaken or they've improved since last year when I was looking to buy!

Monoprice is usually good...