There are two elements involved here: edge engagement and edge hold.
"Short swing" has not been a "current" turn here in the USA in many years. I think of it as an edge set to an up unweight with edge change and direction change in mid air followed by an edge set in the new direction with a steady flow of the upper body down the fall line. I've never seen anyone do this on ice without skidding. Short swing turns require extremely quick edge engagement. When the skis are not pointed in the same direction as the body's direction of travel, this task is virtually impossible on ice on any significant pitch.
Now if you really mean short radius turn, the first thing to consider is that if the turn is shorter than the carving radius of the ski (adjusted for any ski bend occuring) then by definition there must be some skidding to make that short a turn. Assuming you know that and your complaint is about skidding and loss of control above and beyond that; and given that your question specifically refers to steeper terrain and suspects technique issues my guess would be the cause is not rounding out the top of the turn (i.e. engaging the new inside edges above the fall line) due to lack of aggressiveness getting the body to the inside of the new turn above the fall line.
Your skis are going to be a big factor in determining how much edge hold is possible on icy terrain. I'll assume that you know enough to keep your edges sharpened, that you've consciously chosen specific edge bevel angles and that you're not going to change skis to get the optional flex pattern for skiing on firmer snow for your technique. The hardness of the ice is going to determine how much force the surface will support (ok - this is a crude over simplification). Your speed, the steepness of the terrain and the shape of your turn (i.e. how much speed control you want) are going to determine how much total force needs to be distributed over the length of the turn. For a given size turn, your technique is going to determine if the force you are applying to the skis at any point in the turn is going to exceed either what the skis or the surface will support. Changing tactics (e.g. shallower turns more down the fall line, longer radius turns, doing skid to carve vs carve to skid) is another option. The bottom line here is that there are a lot of factors involved. Changing one or more of those factors can get you the results you are looking for. Changing technique is the hardest factor to change. Helping someone change technique without being able to see it is even harder.
Get video! Even video on soft snow would be helpful.