If your shower was tiled in the '60s, it might be a good idea to tear it all out and start over. If there's any sign of moisture intrusion, you're not saving any money by not retiling. There were some good tile jobs back then, but most were not... materials and methods have improved dramatically in the last several years.

There are still hackers out there that will install using old methods or worse, but the options now available to good installers are much better than they used to be.