Certainly sounds like a bad battery, but since the voltage reads only slightly over 12V with the engine running, there is also likely an issue with the charging system, as others have suggested (V reg or generator). The voltage should read about 14-14.5V at fast idle. Spec on the '56 V reg, as a point of reference, was 14.5V. Load test the battery and check or have checked the charging system, as others have said. The charging system isn't ruled out as an issue based on the info you provided.
