Babies have a sucking instinct that needs to be met. This instinct last 4 to 5 months.
You baby doesn't need to nurse every two hours at night time. Instead, you should be holding it to twice a night, nursing her every two hours during the day. You take her to the doctor and have her weighed - if she is gaining weight and has wet diapers, you are doing fine.
Letting her suck on a paci is in my view, better than letting her suck her thumb. It's easier to break the "habit" after the sucking instinct is done if you have a paci to take away, rather than breaking the habit of sucking the thumb.
I know your breast are full at night, but that's because you've been nursing so much at night. You need rest, and your baby needs to start sleeping longer at night too.
Start out nursing at 10:00, and put her down full and groggy, but STILL AWAKE. Let her fall asleep in the crib or bassinet - not on your breast. Keep her awake long enough to fill up - tickling her feet, rubbing her head with a cool wash cloth, pulling on her clothes, etc. Only give one breast so that she will empty it and get that heavier hind milk that takes longer to digest. (If you let her only half empty and give her the other breast, she'll only get the thin front milk that digests really quickly.)
When she wakes later and fusses, put the paci in her mouth. Let her suck on it, and then when she drops it half out of her mouth, take it out of the crib. Only do this for a few months - you want to get her off the paci during the night time as soon as you can.
Look, every baby wakes in the middle of the night. (A lot of us adults do too!) But the point I am trying to make is that you don't have to be the one to always put her back to sleep by putting something in her mouth. You can pat her on the back. You can wait and see if she starts screaming - if she is whimpering, she might just fall back asleep. Don't run into her so quickly, especially if it's not time to nurse her. (Remember, only twice a night! Not every 2 hours!)
Don't pick up your baby - she will not learn to self-soothe if you do. Later when she is bigger, she will not need to nurse in the middle of the night at all. And that's when you should use the Ferber method. Look it up and try it when you are ready to get a full night's sleep. I did that with my babies (I was going back to work) with the full support of my doctor. I HAD to get sleep or I would have made mistakes with people's money in my job.
Good luck,
Dawn