Every driver that lives in an area with snow feels your pain. I usually get a seasonal car wash pass good for one car wash per day and split it between two-three people. That way you can bring them in to the touch less car wash to get most of that salt off.
Good plan. Always try to get all the salt off. When it's super cold and dry, I don't worry about it as much, because as long as the salt is dry it won't cause problems. But if it's damp out, it's time to wash the vehicle really well.
The salt does its worst damage when it's damp. The corrosion process is a result of the salt ionizing in water. Salt's made of sodium and chloride. In water, those two elements break apart into separate atoms. The chlorine is a very powerful reagent which then bonds to metals nearby. Chlorine will even corrode stainless steel. And just getting salt damp will be enough to start the ionization.
This is why you can use a metal salt shaker for years as long as it doesn't get moisture into it. You just put some rice grains into the shaker along with the salt. The rice sucks up any moisture. Uncooked instant or minute rice works the best for this. It makes a good, cheap, and safe dessicant around food.
I have a friend with a pool and he keeps his pool chemicals in his garage on an open shelf near his vehicles. The chlorine tablets for his pool constantly give off small amounts of chlorine gas. Metal items sitting nearby have corroded really badly from it. Tools, brackets holding up the shelves, etc. He thought he had a water problem in his garage until he talked to me and I pointed out the chlorine. Now he keeps it in a plastic patio/deck box near his pool.
I feel your pain, I have a monthly pass to a multi chain car wash center, which I drive my truck thru often during the winter months and then wipe out the door tracks afterwards.
But mud flaps would help too.