NEVER ever hold a piece on a TS with just your hands. Use a crosscut sled, miter gauge.
I agree with using a cross-cut sled. I also recommend that you get a table, sawhorse, something equal to the height of your TS to add for support of the piece your cutting.
After rereading your original post, check your blade that's it's parallel to the miter slots. This can be done with an adjustable square. Your fence should also be parallel to the miter slots. Also, check your miter gauge. I suspect you're experiencing several things happening at once to cause your problem.
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