For some of us, getting low is a matter of strength as much as habit. Try it barefoot on the floor - maybe you don't have enough strength to do deep (e.g., 90 degree) single leg knee bends. (Or maybe you can only do them one way - going down, but haven't the strength to come back up on one leg.)
So squats (deep knee bends), both double foot and (if you can do it without getting your knee to bend sideways, which some of us have trouble with) single leg, might help. If strength is the issue.
At the gym, leg presses (double leg, single leg) are squats, if you start from deeply bent knees, but the machine helps keep your knees in alignment, and it lets you safely start out with less than full body weight if strength is a big issue.
Leg presses on a machine don't fully imitate doing the motion without a machine, so if you can, do deep knee bends without the machine too.
I've seen GOOD figure skaters use leg press machines - though some of them are trying to generate strength to jump higher, rather than to get lower.
Squats and leg presses tire you out too. So maybe don't do it just before or after skating?
I've found that warming up helps me a bit - but, again, don't tire yourself out warming up, just do enough to create a bit of a sweat.
The usual standard now is to say you shouldn't stretch before the main exercise, though you should gradually work yourself up to full current range of motion during the warm-up. They say to do your real stretches after the exercise. But for me personally (very limited flexibility), I find that stretching a bit before hand helps for some skating things - but not for knee bends, where I already have the flexibility to do a full squat. Once you have the flexibility you need for the moves you are doing, getting more flexible can actually be counter-productive. E.g., if you can bend as much as you need to for a shoot-the-duck or sit spin, being more flexible means you need a lot more strength to hold the position.
Shoot the ducks can be practiced off ice, though the balance is different.