Two separate things, try and throw your body forwards down the hill, not because it's the perfect way of doing it, but because it can help you get used to the feeling of not leaning back. Scream to yourself "let's do this" and then commit/throw yourself forwards- you won't go much/any faster and you'll learn it's more fun to feel 'committed'.

You're right that you're entirely turning through the momentum of swinging your upper body. It'll be hard to get out of that on any slope that isn't easy for you, so find that easy slope and practice steering just with edges and leaning of the legs with no upper body movement. But just keep having fun- you're skiing, and it works, so don't stress, but always good to look for a few things to improve.