Getting those kinds of marks out by hand is very very difficult. Its something that you need a machine to accomplish. A buffer from WalMart isn't the kind of machine I'm talking about either. The Porter Cable 7424 is the generally accepted tool of choice for any beginner working on swirl removal by hand. Compounds that contain abrasives are difficult to impossible to break down properly by hand. The speed and heat necessary to remove swirls with those products requires a machine to generate it.
Clay won't do anything for swirls. NXT will help some, but its a very minor swirl remedy. It fills more than it removes and they will still be visible. To remove etched waterspots and swirls I would start with a Porter Cable and Meguiar's #83 (Dual Action Cleaner Polish or DACP for short). That would be my suggestion anyway. There are lots of mid grade polishing compounds that will work. The PC is almost certianly necessary though.