No, to “stay away” is assuming an action that will likely continue in the future. I would “stay away” from people with bad breath, or I would “stay away” from a bees nest. It assumes the action is related to avoiding something (typically relating to the future). To “stand away” you are explaining your position in relation to another object. I would “stand away” from an open window if I were explaining my position to the window. In that scenario, I am explaining where my body is in relation to a window. In a way, they are both usable in similar settings, however they do not mean the same thing. “I stay away from big dogs” means I avoid big dogs now and in the future. “I stand away from big dogs” means I keep physical distance between myself and big dogs if we are near each other.

Stay away - would mean avoid, keep your distance usually, or by habit - "I stay away from trouble". Stand away - physically stand away at a distance right now. Commonly used by the police - "Stand away from the weapon!" In your example, the first (I stay away from those kinds of things) is better and more common.

If you are in the United States, use "I stay away...".