You have received many wonderful answers and i have to echo all of them--It is exhausting, I am right there with you, but stopping breastfeeding will not make the problem go away. It might even create new ones! My daughter was a nursing machine like yours and I never got a good nights sleep. My husband and I really need to stick to our guns with night weaning when we felt she was ready. Pick a strategy and stick with it. Sometimes i was so tired to even try, but I knew we had to. Expect a rought 1-2 weeks of little sleep while you try whatever strategy and remember it will get better. Commit!!!! When I night weaned, my husband would go in and comfort her so she would break that connection. You need a supportive husband for this! If your husband can't for work, or whatever, you just have to make your interactions brief and don't feel bad---you are teaching them how to self soothe and go to sleep on their own--this is a good thing! It might tear you up to hear them cry, but they will be okay. i think once you get a good night's sleep the rest will follow. You will have more energy during the day to pay attention to both kids! That's how I felt! Changing to formula, won't change this or make them sleep better--it can cause tummy trouble. Mine had terrible gas and excema when I did try. horrible, so I went back to breastfeeding.
They just need to break the habits, that's all. Stick to a plan, and remember it will get better....they do sleep! 9 months is also a time when for some reason, things get harder too--teething, separation anxiety, etc.
You might want to start introducing more of a sippy cup esp during the day, to give you a break as well. But this can take a long time, so don't get frustrated.
You are doing a wonderful thing, and you are totally normal for having all these feelings, it really speaks to how much you care about your kids!! Give the sleep thing a try first and see if once you sleep more, the rest falls in place.
Please feel free to email me with any questions too--we went thru a very similar stage with my little one. I firmly believe that breastfeeding moms need all the support they can get since sometimes you feel totally responsible for all their needs and that is exhausting!!