I'm having problems with my the backward inside edge after the FO 3-turn. The check is good right after the turn, but the free leg eventually moves to the side, causing the curve to decrease in size. My coach wants me to fix this, but has not offered me a satisfactory way how. [...] I get the same on-ice problem when (1) I do a mohawk and (2) when I hold a backward inside edge after backward crossovers. However, I can keep it checked for a much longer time. That means there's a problem with the actual 3-turn, but also another problem with holding the back inside edge.
Not necessarily a problem with the turn at all. Have you considered your posture when you're doing these moves?
I have a tendency to let my core go and fall forward after the transition. I don't keep my body in the proper place, which opens the hip, which lets the free leg move out. As my coach explained, your body will go in the direction of the free leg. So if it starts drifting, that's the way you'll move. It all has to do with the transition from one to the other and proper distribution of weight over the skating leg. If you keep it in the center instead of over the skate, this is what happens.
So there are severals parts to the fix that she has me working on:
1) Don't do the 3-turn (or whatever you're working on) like you're going into a jump and keeping the free leg out (as if to pick). Keep it in touch position next to your skating foot. It's slightly in front when you're moving to the front and slightly behind when going backwards.
2) If you must keep your free leg out, remember to point it in the direction you want to go.
3) Look in the direction of travel. For example, in a left FO3, you'd look over your left arm (because it's in front) going into it and over your right arm on the back inside edge.
4) Don't just check with your arms; check with your shoulders. Don't let the arms turn into noodles or the shoulders fall in.
5) Remember your posture. Keep the core tight and stay over your skating foot. Heel, hip, and shoulders should be aligned.
I hope at least some of these tips will help. Good luck!