At 11.5 weeks some LO's do still need to suck for comfort. I would question the hunger at night though, at her age that early feed may be what she needs after all. Have you tried to feed her? it may be that she is waking for that and not the paci. I'd suggest for about 3-4 days you try feeding her first and see if maybe that helps. I say that because one she's still young, and two you said she doesn't always wake for it and sometimes sleeps longer.
If you want to brake the habit not then its a good time to start your LO is still young and will learn other methods of self soothing. There are many sides and positions in the paci vs thumb debate and I don't intent to make this one. I, however, am a big believer in the thumb sucking ring. I sucked my thumb as a baby right through till about 2-3. I dropped the habit on my own according to my mom. I think to many people focus on not having thumb suckers that they inadvertently prevent self soothing or create paci dependency by always giving them a paci rather then letting them settle themselves. Its true with other props as well at times. I understand this desire and I respect it, however, I don't agree with it. I would recommend you try encourageing your LO to find her hand. If she finds her hand she'll learn faster how to sooth herself. If she develops a thumb sucking habit then the key is don't make a big deal of it. the more you bug your child, the more you try to stop them the worse it will get. If you leave them alone most kids drop the habit on their own. Encourage your child to self sooth.
I'm not a huge fan of "cold turkey" myself I personally think its to hard on a LO. I recommend the gentel removal plan. Now you said that its been making it worse. Try leaving her be for a week or so, and then trying again. See if that makes any difference. Your LO may not be ready. You could always try pat/shush before the paci too.
Do you swaddle?? if not try using a swaddle to help her settle. You can use an Aussie swaddle to allow your LO access to her hands.
Also try introducing a Lovie, this will help your LO adjust and give her something else to use for comfort. She may even use that instead of her fist.
You have to find what works for your LO. Continue to listen to her and she'll tell you in her own way and her own time.