Unlike most, I don't have the handgun option but do live in a city where this type of scenario is possible, save for the volcanic ash (we'll call it "generic toxic cloud"). I'd throw the 10/22 and 20-gauge pump in the car with a dry box full of ammo for both.

If I had to abandon the car, I'd put the 20-gauge in pistol grip, 30-inch mode (if it wasn't already) and take 28 shells of #3 buckshot (8 in the tube and 20 in my BOB). I would likely slip the shotgun into a gun sock and strap it to the side of my pack or tuck it inside with only a bit poking out. Though less than ideal for hunting, it would probably be needed more for defensive purposes in the described scenario.