I am going to start out by saying that my advice is mostly for powerlifting, and general fitness/bodybuilding.
First, I'd recommend taking a lower bar position. Getting doubled over in a squat is most common for high bar because of the longer leverage between your back and the bar (it is further away), and because their is no "meat" on your traps like there is on your rear delts that keeps the bar nice and snug.
I'd also recommend widening out your stance a little bit. Partially because of taking the bar lower, and partially because getting pitched forward normally comes down to your bar path being off, your weight distribution being off, or your knees locking out faster than your hips. Taking your stance out some helps because your hips may not be built in a way that you can squat with a proper bar path with a close stance. It also helps because it sounds like either your ankle mobility is lacking, or you are staying on the balls of your feet too much, which are both helped by the wider stance and lower bar position (you naturally sit back more which makes it harder to get on the balls of your feet, and it requires less ankle mobility). You also, strangely enough, stay more upright with a wide stance.
If you are absolutely dead set on keeping a close stance and high bar position (you are kind of forced to do that if you do oly lifting, because staying as upright as possible is the name of the game at times), then work on your ankle mobility, and buy a pair of olympic weight lifting shoes. I honestly don't suggest staying this route unless you absolutely have to high bar squat if you are already complaining of knee pain. Make sure that you also stay on your heels, and you are breaking at the hips and knees at the same time. Hips then knees if you do low bar. You never want to break with the knees first.
As far as exercises to work on bracing and good posture: front squats are a given, good mornings/squat good mornings are also very good. Good mornings work on making you brace your spine, and really beat up your abs/erector. Squat good mornings are also really really good for getting pitched forward, you basically perform a good morning, then squat down and squat back up. It is an overload exercise, and it really emphasizes on making sure everything from your neck to your ankle is tight.
Tl;dr:
take the bar lower, and widen out your stance.
You may be on the balls of your feet, and there is a 99.9% chance that you don't have enough ankle mobility to high bar squat.
Buy some oly shoes or other heeled shoes if you have to keep high bar squatting, but otherwise just change up your form.
Front squats, good mornings, and squat good mornings are really good for correcting your posture and lack of tightness, do them.