Megan, you are correct. The morse call sign was a "shorthand" identifier. The aircraft name would be used for voice transmission so that all listeners, some or even many of whom would not have the five letter group decode that could identify the origin of the transmission - useful, for instance, in an emergency.
Just to clarify my previous post, once Pan American started to use the PA suffix to its registrations, and before flight numbers where used, the three numerals of the registration were used, thus N705PA, a 707, would call as Jet Clipper 705 in 1960 whereas N751PA, a DC7C at the time would call as Clipper 751.