Prep before breaking down chicken
Unwrapping the chicken and leaving it uncovered in the refrigerator for a few hours should dry it out a lot. You may want to sponge out the cavity with a paper towel before doing this. If possible, support the chicken on a wire rack above a plate in the refrigerator, rather than placing it directly on the plate. Otherwise, put the chicken breast-down on the plate (so less of it is in contact).

I'm not sure what the effect of dry brining on the saltiness of stock would be, but it's unlikely to be wildly different than the effect on the meat. If you're worried, though, you should break it down before dry brining, and don't salt the back.