Physics:
The faster you go, the further you can lean in. The further you want to lean, the faster you need to go.
It probably therefore requires a steeper slope than where you're carving now. It will likely require the steepest groomed run you can find. But don't go straight there just yet, work your way up to it.
Procedure:
You're going to traverse at a moderate speed, crouch, lean onto your downhill edge, and extend your legs as you go around. It's going to feel like falling to the inside and letting the board arc around and catch you before you hit the snow. As you exit the turn, the fact that it's a steep slope works in your favor - you could be 80 degrees to the slope but only 50 degrees from vertical. You should exit with your body straight, then compress/crouch to get ready for the next one.
Technique:
I ride hard boots so I'm not the best person to give advice to a soft boot rider.
But, try to hold your rear hand over the nose and keep your butt over the tail.
Try going progressively faster, on progressively steeper runs.
Push and pull.
Be crouched between turns, extend during the first half of the carve, compress during the second half. The steeper the slope and the further you lean, the more helpful it gets.
The reasons that it's so helpful are subtle but real. When initiate a carve on a steep hill, you're engaging the downhill edge, and your center-of-mass starts falling down, so your edge pressure is lower. Extending your legs helps with that.
When you're in the middle of a carved turn on a steep hill, you accelerate. Compressing your legs in the second half of the turn helps to absorb some of that speed, so that you can return to a reasonable speed before you start your next carve.
Snowboard:
If you haven't tried some different sidecut radius boards, you probably don't know what you're missing. Soft booters, especially. It's easy to find alpine boards from 10 to 20 meters but the soft boot world rarely goes above 10m, and 10 is the MINIMUM for fun carving IMO. My shortest is 11.5. I mostly ride 13, and 15-16m is where it starts feeling a little bit too large for my taste. But I still ride those 15+ boards a couple times a year because it's a fun change of pace.
If you've never carved with a sidecut radius over 10 meters, you owe it to yourself to try something in the 12+ range. It'll be harder at low speeds, but it'll be easier to ride at high speeds.
You might find that your old board feels too twitchy when you go back to it. You might find that after you get confident with the speeds required by a larger sidecut, your old board doesn't even function properly until you slow down. You might find that your old board was holding you back. But eventually you'll find that you can do anything on any board, it's just easier on some boards than on others.
Boots:
I'm a big fan of hard boots. Obviously this stuff is possible in soft boots, though.
I avoided hard boots for years, because I assumed that hard boots were just for racers. I also enjoy hitting jumps in the terrain park, riding moguls, weaving through trees, riding switch, etc, etc, and I was worried about losing versatility. Those fears were unfounded.
I put hard boots on a typical 25cm-wide all-mountain board, then went 23cm wide... 21cm wide... 19cm wide... I didn't feel constrained until 17cm wide, so I stayed at 19. Every step down in width took a day to adjust, and it always just felt better. Riding switch never got harder - it actually got a little easier at 19cm, for no apparent reason.
But if you want to stay with soft boots, stay with soft boots.
Setup:
Your binding angles need to be steep enough to prevent your toes and heels from dragging in the snow when the board is tilted 90 degrees.
I believe pretty strongly that the middle of your stance should be at the same place as the middle of your effective edge. Most boards have the inserts set back an inch or two, which is good for powder but bad for carving.
The rest is up to you.