How can I replace electrical boxes with the least amount of collateral damage?
The answer is: Don't.

Outlets are black, beige or white, and rarely dark brown or gray. Beige came into style by the 60s and white by the 90s. Yours are probably beige. 

Leave the old receptacle boxes alone. They ARE grounded, there's nothing wrong with them, they obviously do have enough wire to reach the old receptacles.  That's all you need. 

"Modern" receptacles are the same size.  Just change your receptacles from whatever's in there now, to a quality $3-4 each midgrade receptacle. If replacing backstabs, get receptacles with screw-and-clamp terminals - the wires poke in the back, but clamp by tightening down the screw terminals. Slick. 

If the wires are ALL 12 AWG and the breaker is 20 amps, you can use 15/20A receptacles which have a "T" shape on the neutral slot.  

If you had "back stab" types before, don't cut them off because you'll lose precious wire length.  Instead, pull steadily on the wire while twisting the receptacle back and forth (not enough to deform the wire), and the wire will ease out.  It will have scratches but will work fine with screw-and-clamp or side terminal screws.  Don't use it again on a "back stab", in fact never use backstabs ever because they stink.  Also don't pry it out with a wire nipper because you'll nick the wire. 

GFCI

Ground faults are current following abnormal paths, such as through a human who is being electrocuted.  Mainly, they protect people and pets from shock.   Usual cause: bathroom mistake or defective plug-in appliance. 

If you want GFCI, remember the first rule: Only the first GFCI device needs to be GFCI. Everything downstream can be plain old outlets, and they inherit GFCI protection because they are fed off the "load" terminals of the first GFCI.  That's how the GFCI can be a circuit breaker.  It can also be a new receptacle you insert at the front of the chain, for instance 1 foot from the service panel.  I do that all the time.  GFCI receptacles are cheaper than breakers. 

Honestly it depends what threats you're worried about, but if you're distrust your old wiring, or want to follow the latest code, I'd consider AFCI instead, or both. 

AFCI

Arc faults are loose connections, broken wire etc. causing current to leap across a gap.  This makes a lot of heat and starts fires.  If you've heard the unique "sizzle", it also sounds like that on the wires.  AFCI's have a computer inside which listens for that peculiar electrical noise.  Usual cause: defective in-wall wiring or receptacles; sometimes bad appliance. 

Generally the cheapest way to get AF protection is with an AFCI circuit breaker. 

Combo AFCI+GFCI

You can also get combo AFCI/GFCI breakers, but they are expensive.  Watch for ones which will indicate why they tripped, otherwise you'll go bonkers. 

Another way to get combo AFCI is to use an AFCI breaker, but a GFCI receptacle in the first point in the chain.  This makes it more obvious what tripped.