Can a slab under load be concrete jacked or otherwise supported without taking down the wall?
Here in my town, I have seen foundation repair that is great for loss of substructure support, such as soil compaction or erosion.  What they do is bore a hole in the slab, and inject expanding urathane foam into the slab underside, as they watch the slab level and cracks.  What happens is under thousands of pounds of pressure, the foam replaces missing soil, and litterally lifts the sagging slab. Since the foam is like a liquid, it flows to where it is needed, and supports very evenly.  Once the engineer is satisfied he has the right amount of list, they turn the machine off, and shut the valve, but leave the fitting connected.  They wait for the foam to harden, before they disconnect.  At that point, the repair is pretty much permanent, at least until more erosion or settling causes the need to repeat.

As long as there isn't a place for the foam to escape, this works great.  If there is a sinkhole, or an opening that lets foam out, they may have to inject, pause for a day, and inject again...  till they get the needed results.  Obviously this is for a slab on or in the soil.  For slabs poured over basements, screw jacks and perhaps beams are installed in the basement.

But this is a job for a foundation repair expert.  This is one of those areas a do-it-yourselfer can do more harm than good.  A foundation repair company will have insurance for liability, in case something goes wrong that is his fault.  If you do it yourself, you could make things worse, and there is little limit to what worse means when it comes to supporting a house on a failing slab.