Your wall assembly and shower base way well be saturated completely with moisture and shower goo. A properly constructed shower base will drain water and no standing water will be left in your assembly. A poorly built shower will maintain 1/8" - 1/2" if not more standing water that will wick up the backer board and breath through the entire floor and the bottom 4-18" of your backer board.
It is common practice to lap the backer board over the shower's membrane. If drywall or mastic was used durning construction it could be covered with mold.
If the grout is failing it could be a sign that the bottom plates are swelling and the shower is getting worse. I would see if you can inspect the method of construction and perhaps remove a tile carefully to inspect. Remove a tile not from the bottom but rather one or two courses above the weak area. If you can upload a picture I can draw an arrow on which one or maybe two tiles we would remove for an inspection.
Use a mask when working around the shower. I often get sick shortly after a demo involving large amounts of mold.
Air movement will help dry the shower faster. Consider using a fan to help and dry the shower out.
We use a heavy duty tile and grout cleaner here on nasty showers. Some elbow grease and careful scrubbing can clean a shower up nicely. Try this approach first and if the smell or mold comes back quickly consider inspecting further like I described above.