This is one that has stumped me before. I've mostly gotten past it, but I definitely relate to your issue, Pete. Don't blame the skis; they're just trying to follow instructions, but are probably getting mixed signals from you.
I like epic's posts in this thread, suggesting that less edge angle can be part of the remedy. When this happens to me, I'm generally in a no-man's land between edge lock and skid. The tails of my skis are taking a wider line than the tips, and I'm a little bit on the defensive, looking for speed control, clutching too late at higher edge angles. The immediate band-aid solution, once I'm in that spot, just to save that one turn, is to soften the feet, pull them back, and let the whole ski drift more. Yes, this is counter-intuitive and does not come easily. But heel pressure plus suddenly higher edge angle late in the turn is death, unless you doing the racer-boy thing with significant skill, which I doubt is what you're shooting for on the terrain you describe.
Can you link edge-locked carves on an easy green? Yes? Okay, then you understand tail-follows-tip. But temporarily forget the edge-locked part. From a chatter-prevention point of view, I find it helpful to let go of that altogether on steep terrain. Practice turns where you are on the new edges very early, as on your easy run, but dial back the the edge angles. Pretend you've just enjoyed a nice cup of green tea. You have all the time in the world, and nothing to prove to anyone (including yourself). Follow a similar tails-follows-tip trajectory to the one you'd have in an edge-locked carve, but keep the angles fairly low so that you are drifting a little bit off your line the whole way around, with the whole ski, not just the tail. Stay on this drifting / brushing all the way around, not forgetting counter, to the point where you are not going any faster than you were at the beginning. If you're doing this right, the comforting feeling of speed control at your fingertips throughout the entire turn (not just the the lower belly) should give you the confidence not to over-edge during the last phase. Now release the edges downhill and start over.
(If you get to the point where you can master this on a given slope, you can play - if you want, but not if you don't - with more inside-ski and -leg action as the turn progresses, so that each turn starts with some brushing and ends with an edge-locked swoop into the transition. In order to avoid the chatter in that experiment, the edge has to be locked roughly by the time your skis are headed down the fall line, or shortly thereafter. After that, for most of us, there is too much pressure to get onto a locked edge without ugly abruptness at best, or chatter at worst.)
Hope that helps in some way. If not, move on, nothing to see here.