It's normal. She just needs to suck. It's a strong instinct that actually starts to wane around 5 months.
Like you, I wanted my kids to take the paci. It's far easier to take a paci away from a baby than to get their thumb out of her mouth.
It's not really a "habit" until around 5 to 6 months old. After that, yes, it's a habit.
This is my suggestion, because I have a pretty tough opinion about pacifiers and thumbs being sucked on:
Buy several types of pacis and keep trying to get her to take one of them. I did this with my second, who was picky-picky about not wanting a paci. I couldn't take him sucking on my nipples as a paci - he didn't want more milk - he just needed to suck. And I wasn't letting him near that thumb. He finally took one of the pacis. (My first didn't care what paci I put in his mouth - he was easier.)
At night time, I put them both down with the paci in their mouths after nursing. They were still awake when I put them down (I always did that because I wanted them to learn to fall asleep with me holding them.) After they dropped off, they relaxed their hold on the paci. I took it out of the crib. No more giving it back to them for the rest of the night. I let them suck on it while they were awake.
The reason is that I didn't want to have to keep getting up all night to put it back in their mouths everytime they woke a little and wanted to suck on it. If they got used to not sucking during the night, they woke up less. (Only to nurse.)
At 5 months, I took away the paci for good. Yes, he cried and whined for a few days, but that was it. With my second, we were in London when he was 4 months and his paci broke. No paci in all of England was going to do for him - he wouldn't take a one! That was it for the paci! I was sad about that - I really could have used a paci for the 2 weeks we were abroad, but there ya go! He was fussy for a few days, but that was it.
I never used it during the night, so when I took it away, it didn't change their night routine.
Continue to put the long sleeves on her hand so that she won't actually get her fingers in her mouth. And try several pacis - just keep giving them to her instead of letting her have her hand. At 5 months, no more pacis.
Dawn