Get Thee Onto Snow Blades!
If you try to force your hips into a low position, it's like trying to push rope. You need to let your hips go low in order to stay in balance. To get low hips to happen naturally in a turn, you need to have high speed relative to your turn radius. You're going to need to really carve your turns. You need high edge angles and wide seperation between the feet. You can do all this on skis. Rick's Gorilla Turns and Pierre's comments are spot on, but I've got an easier approach for you.
Try snow blades. Snow blades force you to get your balance centered through negative feedback. Try skiing too far back on blades and you will end up with a sore butt. Snow blades force you onto higher edge angles, or else. Try skiing blades flat on the snow and they will wobble your quads into jelly. Rick's Gorilla Turns will be far easier to do on blades. The naturally shorter turn radius of snow blades means that you need to have less speed to have the same force required to get your hips low. Or travelling at the same speeds you will get your hips lower than you would on skis. Once you've mastered your fore/aft balance on blades, you simply work either on tipping your feet/knees more or widening your stance farther apart (or both together, whatever works for you) as a focus to get tighter more forceful turns to happen. The idea here is that there is not much to think about and BOOM (sorry John Madden) it happens.
Once you can do this on blades, it is easy to transfer this back to your skiing. Most people I've worked with though, have discovered that they really have to "turn up the volume" on their skiing to do this. After locking in the moves on blades, they can do it. But the difference between their new and old skiing is a shock to them. If you ask them what the difference is, they'll say their new turns have a lot more speed and energy in them. But if I had told them to just get higher edge angles, wider feet, go faster, add more energy into their turns .... Well, that just does not work nearly as well.
Although I've seen a lot of people pick up blades and just figure this stuff out by themselves, Pierre's suggestion to get a pro to help you is a good one. The people I've had in blade specific lessons have had a hard time believing they could go to higher edge angles/faster speeds. But after I give them a couple of tweaks and then "push" and "pull" them, the learning is greatly accelerated.
So my recommendation is to try blades for an hour or so. If you don't see a huge difference right away, try a lesson. Gator, if this works for you, we want to hear about your toothy smile (would that be a "Gator Grin"?).