The easiest way to trap die is to just put everything in an eval block. Also see die for info about what you can do with $SIG{'__DIE__'}.

exit is a little trickier, but you might be able to set something up in an END block. (I've never tried, myself... I'm not sure what would happen if you did a goto in an END block). I'd think you'd be better off just removing calls to exit, or changing them to die where appropriate, if that's possible.