Apple ID not working for iCloud

I recently had to change my Apple ID and it seems to be working everywhere except.... today I was using my older Mac Pro - the new ID works fine for everything except iCloud. When I go to System Preferences and click on iCloud it comes up with my old Apple ID and, of course, when I put in the password (which hadn't changed) it just gets hung up - spinning cog/wheel. How do I get the old ID off the iCloud app ?