Congrats on a successful quail hunt! Must be a decent year!
Pull the guts out immediately upon recovery but leave the birds fully feathered. Back at the truck I'll dewing, head, leg them plus skin 'em. Don't put them into plastic bags until you can refrigerate or put them on ice cool. If gut shot try to cut out the contaminated stuff...might be a write-off if if too badly mangled. Carry vinyl gloves for the field gutting,...which is as simple as "cracking" them open at the butt with your thumbs, them pulling out the innards.
Also, maybe open up your choke choice to skeet as well if using a tighter choke Or use larger shot in the future? I always liked skeet and IC with 7/8 oz 7.5's in a 20 ga. Tight chokes and small shot (8 or 9) will usually leave more damaged meat plus more shot in the bird. Or don't shoot them too close as that certainly concentrates the shot charge.