I'd detune the edges completely with a gummi to the point where you can't cut your fingernail on the edge. Edge hold will suck on icy parts but at least you'll know for sure whether the problem is the ski or the tune.
I'm going to offer the completely opposite recommendation. Sometimes, on ice--and "true ice" is always uneven, never completely flat--if you find that you cannot get "off edge", it's because you actually do not yet have a good edge hold; the little ripples on the surface are repeatedly tossing you off the surface. You need some edge hold to be able to push against in order to transfer your weight; otherwise, you become disorganized, like you are floating in space and pushing against nothing, and it will feel like the skis have a "mind of their own".
So, before detuning or any other adjustments, try this: stay in the turn, take your time, wait for that sensation of confidence to kick in as your speed drops a bit. Do not try to flip the skis around or rotate them prematurely. Stay "on the edge" until you feel in control of the edge, then shift your center of gravity over; you will know when, because suddenly you will feel a lightness or an easy, natural bounce. Do not try to turn until you have this cue; on ice, ironically, this just takes longer.
In other words, it may not be the skis that are refusing to get off the edge, but your instinct warning you that you have nothing to pivot against, yet. It is not the skis rotating or catching randomly, but your legs instinctively searching for that sensation of security, and not giving the skis enough pressure to edge with.
Hope this makes sense.
If this works, if anything, what you need is a sharper edge bevel, maybe 2º or 3º (I don't know what the default is on your model). I would not ever go on East-Coast ice (or even West Coast hard pack) with the edges all detuned.
The skis seemed to have mind of their own, particularly the right-labeled ski. I switched them back and forth and found that didn't help so I was able to rule out an issue with my boots.
In my mind, this rules out the skis as the problem—it is unlikely that both skis have the exact same tuning problem—and maybe points to an asymmetry in the strength of your legs. Your right leg is either weaker, or you are not using it as confidently. I'm ruling out boot problems, since you are fine on other surfaces, including crud and powder, which requires just as much force transfer and control.
Added: Thrilluvits apparently have a much smaller sidecut radius than the Kanjos, and so are better adapted to harder slow. Kanjos are rockered on both ends, too, more suited for softer snow, louder and more squirrelly on harder snow. Kanjos definitely needs different technique on ice, outlined above; one could say don't bother with them on ice, as they will be more tiring on ice, but they are more versatile on softer snow.