I never let my kids cry it out, however I also knew letting them fuss for a bit wasn't going to hurt them. It won't. To be honest giving her the gift of learning to sleep well, self soothe is priceless for her. At 9 mos she should be eating less at night, eating good solids during the day, unless she is going through a growth spurt. Growth spurts cause increased appetites.
She could be cutting teeth which will totally destroy sleep patterns. Signs of cutting teeth are swollen red gums, slobbering a lot and loss of apetite or needing to nurse for comfort. Nursing can help relieve some teething pain.
If she is teething, you can give her children's motrin, or tylenol and teething tablets.
The Teeth move and grow mostly at night so that is where the pressure and pain come in. Babies can feel teething pain before a tooth ever appears.
If she is nursing for comfort not necessity of hunger then she will use you as a human pacifier. Do not assume because she wakes she needs to eat either. That isn't good for her and it won't end anytime soon I am afraid. What you want is for her to fall asleep on her own without crying and that rarely if ever happens after habits are formed. It just takes you letting her fuss for increments of time, don't pick her up, but go rub her back and reassure her. If she is conditioned to know you will nurse for comfort or rock her to sleep how is she going to teach herself to fall back to sleep?
I used to rock my babies to the point of really being drowsy, lay them down, rub their back and talk softly to them and walk out of the room putting on a soft music box. I used a fan for white noise that they still will crash out with. If they fussed I gave it a few minutes, if it was a full blown screaming cry I of course scooped them up, comforted them but laid them back down awake. Never feed when you know she just ate to comfort her. You sound exhausted and if you want true change you have to step out of your comfort zone of letting her cry for a bit, or seriously it won't get easier.
Fans now are said to help reduce the risk of SIDS
If she is sleeping in the same room as you, she can smell you and hear you breathing and her sleep will be disruptive. She needs her own room. I got great baby monitor and put my kids in their own cribs out of my room and they slept a lot better. I could hear anything on the monitor and they were only a few steps away.
DO not nurse her to sleep, nurse her and lay her down awake.
It is NOT being cruel, she will not be in therapy. Her sleep is essential right now with her growing and changing so much, giving her that is much more important then having to worry about if she cries she will be damaged.
Let her fuss. You don't have to let her full blown scream and cry but don't jump at the first fuss or pick her up with every cry either. I did the 10 minute rule for my kids. When napping put her in her own room in her crib, put music or white noise on and make sure the room is darkened and walk away.