As we work on a room, we are replacing the outlets. We are using Cooper tamper-resistant (TR) outlets. These outlets have little shutters over the holes, to resist foreign objects, and only allow plugs to be inserted, in theory. However, little effort is needed to overcome the mechanism, so emphasis on "resistant." They certainly are not tamper proof.
To make up for the overly recessed boxes, and over sized drywall hole, we put nylon spacers on the outlet's screws. Put the fiber washer on the outside to hold the spacer on the outlet.
To cut down on the drafts, we used Great Stuff Window and Door spray foam. Just a bit to fill the space between the box and drywall, and, when dry, trim excessive flush with a razor blade. A little into box holes should help, but do not fill the box. In the picture below, it looks like a lot of foam, but the gap between the boxes and drywall is really that large.
Be careful not to get the foam on anything you don't want it to stick to. It can be cleaned up with acetone, but so can dried latex paint. A single use is implied in the directions, but with a little acetone dripped into the nozzle and applicator, it's foam can be cleaned out while wet. Don't let the acetone sit, put a few drops in and then dump the solution out, especially from the nozzle in case the acetone could damage it.
The pre-cut foam pads that go under the outlet covers may help but are too small. Regardless, the spray foam is likely to be better at insulating and stopping air movement.
None of this is advice, do what is best for you, seek professional help if uncertain.
