They way you're doing it shouldn't cause the rust you're seeing, unless you are going so slow that it's giving the barrel time to rust. I doubt that's the case.
I'm wondering if the rust is forming between the time you last fired, and the time you start cleaning the rifle, unseen and under the fouling. The first two pours of water wouldn't get rid of any rust already in the bore, and the brushing could be churning it up some, making it look worse than it is.
Virginia is a pretty humid place, and if you're using spit or water to swab the bore, you could be leaving the range with a moist, cruddy bore. Now, if you zip up the gun in a warm guncase for the trip home, it could make this situation worse, just right for a little superficial rust to grow until you clean it.
Or, you could just be waiting too long before you clean it, allowing time for rust to grow.
If this sounds like it may be what's happening, you could dry out the bore a little better before going home. Or if you're waiting a long time before cleaning, try hitting it sooner after shooting. Hope this helps. Bill