I'd be a bit careful with the cotton canvas--you really don't want to get it wet. In very cold, dry conditions it's fairly easy to keep it dry. In North Carolina you're going to be closer to melting temperatures much of the time.
As for managing the sweat, it's all about adjusting your layers. Start cool (as others have mentioned) and adjust. You'll have to do it a lot if you're going up and down hills. Your uphill mode will probably involve no hat or gloves, jacket zippers open, and any vents you have (jacket or pants) open. If you're lucky you won't actually have to remove insulation layers (which requires taking your pack off).
Even the best rain jackets money can buy aren't very breathable. Use them only if you need one to avoid getting wet (from melting snow or rain) or if you are cold (they are good wind breakers).
I don't have a lot of snow shoe experience so I can't offer detailed advice. I have used them to ascend some snowy slopes and they (at least the ones I used; MSR Ascent series if I recall) are very good at that. I own a pair of the Diamond Trekkers modustollens mentioned and have been pleased with them.
Let us know what you end up doing. I'm curious how well the anorak works if you decide to go that route. Just be careful with the cotton!