I meant to reply with my two cents but forgot.
I generally don’t walk the trails, I walk perpendicular to them and step over them. When checking, if I can’t do it from a distance, I do the same thing, always stepping in my own footprints. When I set a snare I stand to the side of the trail. Whenever possible I like to hang my snares just before it snows or while it’s snowing.
Probably a bad idea to wash your Rams with scented dish soap but it doesn’t sound like that’s what the issue is. If you are getting refusals it’s more likely that they saw your snare rather than smell anything. Try setting in bends in their trail if you can rather than setting on a straight stretch where the can see the snare from a distance and blend it in. Catching them by the muzzle is likely that they felt the snare, wouldn’t commit and started to back out when the ram went off. A couple of years ago (not using a ram) I had a snare pulled full length and the noose was the size of the yote’s muzzle. His track in fresh snow showed him backing out of the set. The 10”-12” x 10-12” shouldn’t be an issue.
Scent isn’t a big deal with my farmland coyotes but I do pay attention to it. I boil my snares with baking soda to dull them but I attach them to my extensions using clean, bare hands. Then they just get put in a Rubbermaid container in the cab of the truck. I have one pair of gloves for handling bait and snared coyotes and another clean pair for handling snares. A lot of hanging snares is done with bare hands. Clean gloves go on the dash and dirty gloves go on the floor.
Around here coyotes are territorial and will defend their territory from trespassing packs and transient coyotes. Once I trap the local pack catches get slow and I might catch the odd straggler. When that happens I start a new bait site in a new territory and when that one slows down I move again. Every once in awhile I find a honey hole that keeps on producing but that’s not usually the case. For me, 14 coyotes at one baitsite is a pretty good number before slowing down. You may have trapped all the locals and only catch transients now.
This is just my opinion based on the coyotes in my area and things may be different in yours. Good luck!