@GardenLover
This is not a silly question, of course.

I'm not an expert, probably, but I also cut off my adeniums. I do not process the cutoff, the branch dries up by itself.
Never rot.
I cut the branches differently. Not all at the same time.
I cut the old branches one at a time. I look where the plant has the kidneys. And I'm doing a cut on this place. Then I wait, when there will be new young branches. And after that I cut off another branch. I suppose that the plant must have a balance between the roots and the crown. Especially if you have a hot climate. Maybe I'm wrong, but I think your plant withers because of improper pruning, but not because have not processed the cutting site.