There's plenty of threads about bouncing hands / bouncers so if you run a search you'll see how popular they are
Congratulations on not being one of them anymore
As for the body bouncing, how about trying to keep your feet as much as possible on the floor... as if you were 'gliding' or sliding indeed.
It's easyier to do that if you have a smooth sole (leather or suede - again run a search on shoes if you want more details).
Just try to 'caress' the floor with your feet, i.e. always maintaining contact between your feet (or toes) and the floor. It will be a bit weird at first and people may start to wonder what you're doing but just take this as an exercise until you are used to it. That should stop you from bouncing (difficult to bend your knees at the same time!). Then once this has become like second nature you can relax and 'just dance' and use that 'caressing' technique solely to match something in the music (like when it feels like the music (or just an instrument or vocals) is 'stretching' in time. But that will be another journey in your dancing (look for musicality and 'understanding the music' threads if you're interested).
I hope this doesn't sound too patronising , welcome to the forum.
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