i used a piece of plywood clamped to the door as an outer-guide. Cut a hole in the plywood, the right size (2 1/8) and then clamp it where you want the new hole in the door. Then using it as a guide on the outside of the hole saw, drill on through the door. voila. no additional tools required beyond the 2 1/8 hole saw, and well, a c-clamp which you should have.

This style lockset is known as a mortise lockset.

Dis-assembly and cleaning won't hurt.

While you've got it apart, print the Baldwin diagram below, and note your dimensional differences. Some will be critical (holes for knob and deadbolt), others may be compensated for by strike plate adjustment or wooden block supports (under a shorter mortise body).

Backset (the distance from the door edge to the center of the knob) is a critical dimension that will determine what replacements are available. Modern backsets are usually 2 3/8", many older mortises are 2 1/8 and 2 (the new one below is 2.5)

This page lists some Baldwin mortises, they cover a range of backsets and door thicknesses. Googling "mortise locksets" will net you many choices.