I remember it being a subject of discussion when he was elected and I remember the Rose pronunciation being spread as the correct one of how she said it. It was slightly unusual because there was a famous movie star some years before that who was named Rosalind Russell and she definitely had the Roz pronunciation, as did others named Rosalyn. Keep in mind also that there are regional differences between the North and Northeast and the deep South, where the Carter's are from. To me Rosalyn (with a Roz) sounds like a northern name, but maybe that's just me.
Some "first names" are really two separate first names grouped as one concept, so Sue Ellen can be a first name used by someone who is never known as simply Sue or Ellen. She is Sue Ellen. Sometimes those names can be joined. My feeling with no evidence is that Rosalynn Carter's name comes from this tradition where it's ultimately a combination of Rose and Lynn.