I've read through a ton of information and found 100 different ways to move to a new server where the hostname and IP were changing. The process that ended up taking was this:
Note: We are using a SQL database. Server1 had SEPM 12.1.4 on it. I updated this server to 12.1.5. I then installed a fresh copy of SEPM on Server2. After installation I did the Management Server Configuration Wizard, in which I chose to Add a Management Server. I went through all the steps using a recovery file so that it would automate most of the information, setup the proper SQL connection information and added Server2.
At this point I was able to login to SEPM at Server2's address, go to Admin -> Servers and now see 2 servers underneath Local Site. I then added a Management Server List that included only Server2 and assigned it to all the machines. I then shutdown all symantec services on Server1 and began updating all the clients with the latest client version and made sure they all connect to Server2.
After all the updating, all the clients successfully were on the 12.1.5 client and communicating with Server2. At this point, Server1 is sitting there with its services disabled, however it still appears under Local Site in Admin -> Servers.
Here is the weird part. If I right-click on Server1 and choose Edit Properties, or Delete Selected Server, or Manage Server Certs, it always gives me Server2. It literally changes the window's information from Server1 to Server2 as if Server1 doesn't actually exist. So my issue is that I am unable to remove Server1 and I don't know the proper way to complete this "migration" of servers.
I am pretty lost at this point as to what situation I am in and would appreciate any guidance. Thanks.