Yeah, having had "temporary indoor wading pools" as well I can appreciate what you're going through.
Once the cartridge ammo dries it'll be fine. Brass cartridge ammo around here goes through the washing machine more often than I'd like to admit. Mind you so does pocket change. I REALLY need to learn to check my pockets before laundry day..... So cartridge ammo will be fine once you dry off the outside of the rounds. Do NOT put it into the dryer. Aside from the heat issues the noise will drive you bonkers.
Any cardboard that the rounds and shot shells are in is done. You may as well toss it. Or at least open it, empty out the ammo and set the boxes out to dry if you really must.
Looking at a shot shell i'ts pretty obvious that they have a hole in the middle of the crimp. If they sat submerged then try drying off the outside of one and stand it on end on some paper towel. If the towel is seriously wet after a couple of minutes then the shot at least got wet. But even then the plastic wads could well have sealed out the water from the powder and primer. It's a pain but set 'em out on the crimped end for a day to drain and then tip them over on their side so they can dry out as well as practical. After a week or more try a few out and see if they fire.