I do know that rocking is common among people on the spectrum. Although I haven't heard of sighing as one possible trait, it does sound like one of your issues. And no, you are not crazy, the rocking is just a kind of stimming. I've rocked forward and back, or side to side, but more when younger. I have always been prone to knee bouncing while sitting, and sometimes slowly move a foot up and down repeatedly. I also sometimes twirl utensils or pens. This happens without my realizing it initially, but I will sometimes notice I am doing it. It does no harm, so I don't bother trying to stop it. Besides, every bit of physical movement burns calories, and I need to burn all I can.
Because the sighing is bothering others, I recommend trying to stop it, if you can. As for the rocking, I suggest finding another, less obvious physical stim, as that one bugs your mum. Perhaps the knee bouncing, or better still, foot tapping. Those done with hands tend to be more obvious, and might attract unwanted attention. The exception would be fiddling with a pen or pencil, especially if you keep a doodle pad handy. Doodling is generally considered acceptable, while many other forms of stimming are not. Fiddling with a pen or pencil is usually not taken too badly, either. My utensil twirling has been noticed, but doesn't bother anyone, they just get amused by it. My older sister shreds napkins and tissues without realizing it. She's an NT, but with some Aspie traits.
Hope this helps.
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If a man does not keep pace with his companions, perhaps it is because he hears a different drummer.
Let him step to the music which he hears, however measured, or far away.--Henry David Thoreau