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If you look through my older posts, you'll notice in my swing video that I do not wear golf shoes. My front heel also tends comes up a little bit. I am wondering if wearing golf shoes would help at least plant the ball of my right/front foot? Does anyone have any experience with wearing/not wearing golf shoes and similar problems? What other problems may result from not wearing cleats?

golf shoes may help restrict your foot somewhat, but you will still be lifting it.
you tend to shuffle your left foot a little (reasin below) and the shoes will restrict that also. but if you hitting the ball well, i wouldn't change it. looks aren't everything.
now, said that,
it looks like your stance is a little wide and the backswing is pulling your foot off and then has to re-adjust. bring your left foot in about a shoes width and your foot should roll up on the inside edge with very little, if any, on the toe.