Wanted to know if the wardrobe would allow this- after getting successfully infected, equip a different skin while the infection progresses?

Or would that glitch out my progressing infection or refuse to save?

You can put on a different skin, but you can't bite with the older skin while you have a new one progressing.

- Awesome, time to go play with my skins :D Thanks!

Will that slow the progression of the new disease at all?

No, it shouldn't affect the new strain c: