I cannot see that a scent free detergent would make much of a difference. My washer smells like the detergent that I always use. I cant imagine that smell not getting in anything that is washed in it, same for the dryer.
Maybe if you washed everything in a bucket and let it air dry?
I have known hunters that would hose down their clothes and then hang them on the porch for a week.
But then I hunt the wind and don't worry about scent stuff too much.
"Man is still a hunter, still a simple searcher after meat..." Robert C. Ruark