Further updates for anyone going through a similar issue:
While using “Assign to different site” worked smoothly, we did have some issues with browsers pulling up the old site (even after 48 hours for DNS propagation had passed, though I don’t know what needed to propagate as the DNS was not changed… the URL was already pointing at Wix). Emptying cache & cookies did not solve the issue. We were able to pull up the old site in private browsing windows and on mobile devices that had never accessed the site before.
Further, there was some very weird behavior with the old site vs the new site displaying. For example, I could refresh the browser window (we tried Chrome, Firefox, Edge, and Safari) to force the new site to show, but it would glitch and show the old site again, somewhat randomly.
This topic had the same problem without resolution: How can I make sure my new website appears when users type in my domain, instead of my old website?
I ended up using this embedded code to force a refresh every five seconds: Wix Editor Request: Setting a Page to Refresh Automatically | Help Center | Wix.com it was not a long-term solution as no one could navigate the site as long as it was refreshing. But it did seem to force most of the client’s company to see the new site. Though again, everyone experienced seeing the old site periodically as if through some kind of glitch.
On with Wix chat, they suggested I unpublish the old site, so I did so. This didn’t seem to help anything, even after 48 hours or so. It also caused extra problems as with the old site down, I couldn’t manage the 301 redirects, so we now have Google traffic that is getting 404 pages.
I don’t anticipate anyone being able to help or actually respond here. Just noting for anyone else. If Wix does pay attention to these posts, again, I beg of you to document the process for site redesigns when utilizing a non-Wix domain name already pointed to Wix. And consider building in some DNS reset tools that make the site redesign display more smoothy across the globe.