I want to transfer a genus to a different family. I am thinking the proper way to go about this is to edit the genus and change its parent name, instead of creating a new taxon under the new family and then making a swap. Is this right?
Yup. Under the species page: Curation --> Edit Taxon --> Parent Name box. Then, replace the parent family with the new family (and click Save Changes at the bottom).
You cant do it with a taxon change, only by editing.
Depending on what family it is however even as a curator you may not be able to do this as many groups such as mammals, birds, reptiles, fish just to name some examples are locked and may only be edited by specially designated taxa curators.
If that is the case, you need to flag it for them to do.