I'm not sure what is ultimately causing the mobility in your position. I really looks unnatural.
Do you have the chance to do a lot of non-stirrup work? maybe it's a problem with lower leg and back musculature which makes it difficult for you to maintain a quiet position.
I do have back/ab problems. Even with an instructor watching me do ab exercises like crunches and all different types of ab exercises, making sure my movements are correct I never feel it in my abs, I only feel it in my back.
Have you had the chance to work with different trainers? Maybe it's the teaching style of your current instructor that just doesnt work well with your learning style. It sounds like your instructor is a younger lady/fellow? with not the teaching experience that you might require.
Good for you for working out. I wish more riders would follow that example.