Is it possible to keep the old domain on the site as-is and point the new one to it as well? If so, will it affect the search engine rankings on the existing domain? Thanks in advance!
You can do a domain redirect to redirect user from new domain to old domain. Note the url will change to old one.
If you wish to display the new one, you can use domain mirror to server two domains with the same content.
I was going to recommend Mirroring, but then realized that this person doesn’t seem to be hosted here. Probably spamming the forum.