I'm sorry I haven't gotten back to you sooner; I had computer troubles and mislaid your message in the process.
My first reference to a Morsing is: Aaste Torstensdtr. Morsing who married Christopher Eriksson Glad, a Norwegian (probably from Bergen, though I have to find my original sources). Aaste's grandson was Christopher Giertson Morgenstierne de yngre (ca. 1628 - 1684), so that would place Aaste's life in the 1500s. Aaste was the daughter of Torsten Soeffrensson Morsing and Anna Amundsdtr.
You said there was an island Morsing in the Limfjorden -- is that the same as or different from the island of Mors, which I believe is not far from the town of Thisted. I assume that someone from Mors might be called "Morsing." Coincidentally, another correspondent indicated the island of Kjoes (or Kyx, as he spelled it) also lies in Limfjorden. This is possibly a home of the Tornekrands family, a member of which went to Sogn in the late 1500s.
I don't if any of this has any connection for your Morsings. I hope it does. If this suggests anything further, or if I can supply any more information, please let me know.
Med venlig hilsen,