The key with "attachment parenting" and "the family bed" and this kind of thing is *balance*. I nursed on demand as well, and nursed both of my boys to sleep, but neither of my kids did this rapid sleep-wake cycle. However, when my younger son was about 20 months old, and was *still* waking up 1-3x a night to nurse, I was getting very tired of it, so started refusing to nurse him at night, and then ended up weaning him altogether. It was a rough go for a while, because he didn't understand why all of a sudden I wouldn't let him nurse, but after we got past that rough part, he began sleeping all through the night.
I also noticed sometimes with both of my kids (at 4-6 *weeks* of age with my older son, and somewhere around 7-9 months w/my younger son), that they slept better in their cribs. I think I was waking up my baby with my sleep movements, or I would awake too easily when he stirred but did not wake up. So, although I planned on having him in bed with me, he was transitioning to the crib by 1-2 months old. With my younger son, there were times when he would nurse and then stop, but not go to sleep, and I finally figured out by his movements that he was saying, "Ok, mom, it's time for me to go to my crib so I can fall asleep, 'cause this ain't working any more" -- but there were still plenty of times that he *did* fall asleep nursing, and *did* want to nurse to sleep.
The only way to change a habit is one day at a time. She's in the habit now of expecting you to nurse her at the drop of a hat, and while that was okay for a time, it's now out of balance, to where you're getting exhausted. It's okay to restore some of that balance and tell her no. You're not a bad parent if you do (and in fact, never telling your kids no is not being a good parent -- it's letting the kids be selfish and ride all over you) -- the key again is balance.
You can totally change your bed-time routine, or you can ease into it. You can still nurse her to sleep (lying in bed, or sitting in a rocking chair), and then instead of leaving her in bed, put her in her crib. Since you know she's not nursing out of hunger, I would suggest that if she wakes up in the first couple of hours, that you just pat her gently, let her know you're there, and tell her that she's not going to nurse, but she needs to go back to sleep, and leave the room, going back in in a few minutes. I'm not suggesting "cry it out" (and there's a book called "No Cry Sleep Solutions" that I've seen suggested, but haven't read myself), just suggesting that you gently get her (and you) to change habits so that she knows that she won't get to nurse every half hour, and so that you can get some much-needed sleep.
If you only nurse her every 2-4 hours (whichever you feel comfortable with, and feel free to be flexible), then she'll eventually recognize that there is no point in waking up and getting up since she's not going to get any nursing, and will probably sleep longer. But it will probably get a bit worse before it gets better -- she will protest this change, and demand to nurse, and cry, and you will get less sleep for several nights and perhaps longer. If you stick it out, there will be a time when it ends. But if you keep doing what you've been doing, you're going to keep getting what you've been getting.
Part of your baby's attachment may simply be personality. Both of my boys are "mama's boys" but my younger one is **much** more so. He's now over two years old, and although he's perfectly happy with his dad or with my mom, if I'm in the room, he wants to be with me, and he follows me around like a shadow a lot of the time (if he's not playing with his big brother, which does happen a lot of time). And he's always been more clingy. When I leave him with my mom or husband (which isn't often), he's usually just fine and acts like he doesn't even know I'm gone, but if I'm in the house, then he just wants me. So, you may just need to leave her with your husband for an hour or two on weekends and go to the grocery store, and see how she does -- probably just fine.
I think your solution of putting her into her crib is a great one -- and the only way I know of to do it, is just to do it. She'll get used to it, and then things will be much better.