I couldn't find a way to move a board I had made to an organization I created later. So I copied it over. Now does the same board just show up in both places, or are they now 2 different boards? If I have copied it over to an org, can I safely remove it from "my boards"?
"My Boards" is the catch-all for any board that doesn't belong to an organization. A board that appears in both "My Boards" and your organization's area are two different boards. You can confirm this by comparing the URLs; the letters and numbers following "https://trello.com/b/" in the URL is the unique ID for that board and in your case they should be different. Feel free to close the board that you aren't using, or rename it and use it for something else!
To change the organization a board belongs to: Show Menu > Settings > Change Team