What others have said here is good advice, sugarluver. Here's how I summarize this, with some additional commentary.
It seems that you are right-foot dominant (everybody has a dominant side, and it's a very tough thing to overcome as your skiing ability improves), and this makes you rely on the right leg far too much when you are making right turns. Overcoming the reflexive tendency to rely on a dominant leg is very, very tough: work in small steps, working from gentle terrain up to the steeps. Having a lesson with video analysis would likely help you quite a bit in visualizing the problem - save some shekels and spring for a private or small-group lesson one of these days, it's worth the investment.
There could also be some alignment problems (there likely are, in fact), and I'd definitely pay a visit to Steve at Christy Sports to have your alignment assessed. Thing is, if your boots aren't giving you the proper support and alignment, the best skis in the world will only be of limited help. The alignment adjustment will also help you in achieving the edge angles you need to carve on the steep. If you have proper leg and foot alignment, you'll be more trusting of your left leg and foot, and won't feel the need to lean on your right leg to feel stable. Again, this brings us back to the foot dominance issue - but having a properly-aligned boot setup is key.
And there are other things at play here. I grew up skiing in Utah, and the soft snow of the west allows for a lot of technical "variation" that wouldn't pass muster on the hardpack and ice of the east (or on a prepped race course). Your mentioning issues on the steep suggests that you aren't getting good edge angles in the snow, and that can be caused by lack of hip angulation, leaning into the slope, rotating the shoulders by driving the outside hand forward while letting the inside hand drop back (this causes most skiers to lose hip angulation as they rotate, which causes skidding as the edge angles decrease), etc. In the powder, it's possible to get away with some of these moves and see little-to-no detrimental effect on your skiing. But when you move to a more packed situation, or something steeper, you see the skidding, chattering and bouncing described in your initial post.
One last tip: make sure your fitness level is good. While the alignment issues will create a more stable platform for you, having good core and leg strength (as well as good flexibility) helps immeasurably in being able to have stability.
Whatever the case: have fun! Living in the Wasatch is quite the treat.