Outlet in laundry room is dead
Good diagnostic info!

What happened: there is a short in either the cable between outlet A and B, or in the receptacle at outlet B. Have you tested connectivity/resistance between the poles on outlet B? You might find the short. If this setup was kinda sketchy in the first place, other electrical work may have disturbed it just enough to cause a short; it's happened to me when working around around 75-year-old wiring.

If the current wiring was fished through the walls, it is possible to use it to pull new wiring. However, if it was installed while the walls were open, it will probably be stapled in. Horizontal runs are also not likely to be fished. So depending on the location of outlet B, it may be easier to run new wiring (e.g. down the wall into your basement) than replace the existing.