I live in a region where they put salt on the roads during winter, too. Doesn't it leave such a lovely film over everything? As you say, very gritty and harsh on the car's underside and paintwork.
What I use to "de-toxify" the salts on the car: a garden-sprayer (attachment for the hose), with the
StarBrite Salt Off product. Diluted at ~1:15 or 1:20, it works rather well to neutralize and clean off the salt residues on a car.
For drying, I've got a few of something called The Big Blue Towel. Purchased them years ago and they no longer seem to be available, though AutoGeek does have their
Cobra Guzzler Hybrid towel for drying that looks nearly the same. After a wash, I go over the paint surface and windows with a couple of these, and use another for daubing the little drips and whatnot that keep coming off the tailgate, bottom of the doors, off the grille, etc. (Funny how many drips keep coming out of those places, after a wash.) Soaks up a great deal of water, and they're soft enough for gentle use around the paintwork. Works even better, if the paintwork has a hydrophobic type coating and/or the soap used has a hydrophobic property.