It occurs to me that there might be a design solution that could address these problems. To my mind, the key is to have walls that move down (or extend sideways) continuously, and which are able to interlock to withstand the pressure of a cave-in.
The diagram below illustrates walls that interlock (and could actually be bolted in place) sitting inside the existing walls of the tunnel. These would be continuously advanced as the digging continues (for vertical shafts the force of gravity would be sufficient, but for horizontal shafts some other force, e.g. a spring mechanism, would be required). Once the new section of wall has cleared the old section, it is expanded and corner pieces are added to form a rigid structure (probably bolted together).
This approach should also work with other shapes. For example, to achieve a cylindrical-like tunnel, four quadrants of a circle would be expanded with straight sections.
Additionally, for horizontal shafts a rhomboid type structure may be preferable as the risk of ceiling cave-in is more significant that side wall cave in.