I also have cold well water. So... I used several buckets of warm water, Excaliber, and clean rags (that I threw away afterwards). For rinsing, a large syringe works great. Just fill it with clean warm water from the bucket, and use your hand to position it up the sheath and squirt gently.
Soda didn't mind at all and he's pretty touchy. I have to say I've noticed since I've cleaned it he's been.. er... "playing" a lot more....
As I said above. Move slowly, be gentle, and watch your horse's ears/attitude. Don't try to rush it and if you aren't sure how he's going to react start with water/K-Y or something else that doesn't necessarily have to be well rinsed.
Or just find a vet that will do it.