Installed Plex on New Computer and Keep getting Run Media Server when trying to install new dir

That suggests that you have claimed the new server using that account.
You need to “unclaim” the server.
Log in using the account that allows you to access the server settings.
Go to Settings - [server] - General
and press on the Remove button.
Then log out from the web app as well. (click on the user avatar in the upper right corner and use “Sign Out”)
Now stop and restart your Plex server using its Plex task tray icon.
Now claim the server into your Plex Pass account by following this procedure: New Claiming Requirement for Windows