Ashtabula bottom bracket pressed cups loosening problem
 

Yes - that. The pressed cups are coming loose despite tightening the locknut et al. to the point of unrideability. That is, they're loose inside the shell. Is there any way to troubleshoot this, or do I need to replace the cups?

Most likely the shell is slopped out. Shim it, preferably with brass shim stock, but soda can will work. Get it so the cup is quite snug starting into the shell, then take a center punch and make lots of small pips around the outside of the cup, and press it in with a vise or a C-clamp, after smearing it with LocTite.