How do I correct my position?

IMO the only way to resolve this problem would be to take up some other form of exercise that works on your posture, yoga, ballet, pilates(pilates is more gentle and your probably less likely to injure yourself, the others are pretty intense even though from the outside they look/seem quite mild)

Bareback riding or without stirrups/reins can help also.

Also I know a few trainers/riders that swear by having massage therapies for them and their horse and working on their alignment, because if you have unknown injuries you can naturally be holding yourself weird without knowing and then overdevlop certain muscles, so your natural position can then become hunched.

Some would say to focus on core strength, but everything is important in riding, you need good upper strength and lower also, having good core strength doesn't mean much if you have rubber arms and legs.

Also although expensive being checked for any injuries is good.(not an xray due to the radiation but scan or other methods)

I had fallen off a few times and because I just felt bruised I assumed I was okay. I only found out a few years later that I have some quite bad injuries, my muscle had built around them and I aggrevated them doing other exercises. I was even told that with the injuries riding was now too dangerous.

If you're overarching your back, try to stick your legs out in front of you so you are still straight. But take into account it could be difficult depending on the shape of the horse you are riding or the saddle you are riding in. Perhaps ride a different horse or have someone else ride yours to see if it is just you or not. Another beneficial key point is to be sure you have proper equitation everywhere else. By being sure your weight is in your heels, you will ensure that you are sitting on your seat pockets, so to speak. By holding your hands low and following the bit you will ensure you aren't leaning on the horse's mouth and pulling yourself forward. It can never hurt to take up another dicipline too. If all else fails, a riding instructor should be able to point you in a good direction (up). Actually, don't resort to 'all' before seeking a trainer. Keep it smart. Ok, I'm just rambling now. Happy riding!!! ;)

bring your toes up and calves of legs forward, feel the back of the saddle on your back hip pockets. Stretch your head neck and back as tall as possible.

The forward seat you are mentioning we used to call the ejection seat because you are getting forward of the horses center of gravity. That seat will teach horses to dump you.

Try to bring your shoulders back and sit like that at every meal, so you get used to it.

sounds like your legs are probably falling back, and I am thinking it might be your saddle.