It has to do with the way you drop from your stance. It's actually a progression of the way you used to drop forward onto both knees with your feet behind you. Now you're getting a leg out to the side, which covers more net. But to do that, you drop with all your weight onto one knee, and end up leaning that way. Technically, you need to learn how to drop straight down by bringing both knees together and driving them into the ice. Of course, that doesn't really help explain HOW to get yourself to do that.
My suggestion would be to force yourself to work on that whenever you can - during warm ups, during stoppages, between periods, open ice, stick and puck, whatever. Even in warmups, you can just stay down in the correct position with your wegiht centered and both knees down. Then get used to moving your head and torso into the shot from that position. Work on using both legs to kick out from the middle, instead of just your left, and do it without leaning to the opposite side as the leg you kick. You can have the exact same warm ups as now if your team is smart enough to not dome you.
Here's a video of Tretiak warming up and all of the exercises he would do. You aren't playing as old of a style as him, especially with the pad stacks and kick saves. But you can see how he is controlling and centering his weight, and then apply that to your technique. The drill he is doing without pads on where he looks like a Bolshevik dancer may very helpful for you to try at home or something. It'll get you to kick with both legs while centering your weight.
I used to drop just like you. Then I went to a coach who had me do these "Tretiak-like" drills. We'd just get in a stance and he'd fire a puck to one side. We just were supposed to drop and make a leg save (modern butterfly, not kick save) to each side and get the stick going that direction. then we got up and he did it to the other side. If you have good teammates, maybe you can work it into the end of warmups. The pendulum drill may help too. Drop to your knees with weight centered. Lift one knee and engage the skate blade, then push hard enough to move two feet. After you push, bring your knees back together, and as you are sliding engage the leading skate blade. Use your momentum to bring that knee up and then push back the other way. Do that a few times back and forth. The trick is to keep your body centered, but your weight over the push and then trailing leg. Then use the momentum to load your weight over the other leg for the push, again keeping your body centered.
Here are a few videos of what I mean. The last is a third by Maria Mountain without gear on that really breaks down the mechanics of it. It's pretty helpful.