This doesn't have to be scary at all. Make a backup with a new name and then use File > Save As to save a COPY of your database. Now you are doubly protected.
Map things out on paper the way they should be, using the names and RINs.
Then carefully unlink all the children from the incorrect parents. And relink them to the correct parents. This can be done by having the correct parent in the main position in the FamilyView screen, and clicking Add Child > Select Existing Individual > and use the RIN. (you can use Ctrlt+L or double click in the children space, if you are more comfortable with that choice)
When all is well, make another Backup with a new name, and the job is done.