Pamela,
Glad you're getting there!

That must have been quite a high quality unit for the time, as the Shure M44 is a magnetic cartridge, as opposed to the more usual ceramic/crystal type found on a lot of domestic equipment in the early 1960s.

A replacement for a Shure M44 shouldn't be too much trouble. If all else fails, you could contact Shure directly. I don't have their details to hand, but they're based in Evanston, IL.