Hi there and welcome to BW!
First, you're not crazy... most of us have been exactly where you are. Tired and desperate. You're welcome here.
Second, please know that BW is firmly against any form of CIO alone. We believe it breaks the trust between baby and parent.
It's not unusual for bf babies to nurse more often or longer at night as they digest bm at a quicker rate and get hungry earlier. It depends on each baby and mother of course, but I bf 2x a night until 9 mo or so. I'm not surprised at this age that she is eating eagerly at night... how long ago did the growth spurt start? Usually they last about a week or so. Also, if she's bf one breast at 5, and the other side at 7, she might actually be a little hungry before bed... could you offer 2 or a bottle cap off and see what she does?
Routine- your awake times are slightly longer than average for a 5.5 mo which is around 2-2.5hrs. Did you recently lengthen the AT? She might be accumulating OT if she's not quite ready for 3 hrs between naps... especially if her nights are broken.
Having said that, you very well may have a sleep association. We can address all the other issues, but ultimately, we won't know for sure what's causing the wake ups until you drop nursing/holding to sleep.
Teething- if there are signs of teething such as rosy cheeks, red gums or teeth poking under the gums, I would medicate before BT. I also did dream meds when it was bad... woke him up around 4-6 hrs after the last dose to get him through the morning hours when any discomfort can wake baby (usually between 4-6am).
Congestion- if she does in fact have a cold, I honestly would wait for her to be well before trying these tips, except for the teething medication and reducing the A time, as she could be exhausted from the illness and broken night sleep.
Night wakes- The habitual waking at 10:30/11 and 3 could be due to the habitual feeding, but again... she might be going through a growth spurt still. The 10:30 wake and 5:30 wakes can also indicate overtiredness, in which case I would try to shorten her A time slightly, and aim for an earlier BT, say 7:00 or 7:15.
Naps- you say they're tough.. how so? Does she not go down easily?
To summarize, I would try these things in this order:
1. Rule out illness. If there is an illness, do what you can to maximize sleep for both of you until recovered- offer meds for teeth if sure, and reduce A time, but don't proceed with sleep training or reducing NFs.
2. Rule out teething pain with meds before bed and if bad, another dose at one of the night feeds.
3. Offer both breasts at 7pm or combination of bottle/breast and see if she takes it.
4. Try reducing the awake time starting with the first nap. This is going to be tricky bc with a wake at 5:30, her first nap will be so early. Hopefully, we'll get a wake closer to 6 at which point you can push the first nap later and later. On days where she wakes at 5:30, aim for a 13 hr day... BT by 6:30 if possible. Ideally you want a 12-13 hr day.
5. Continue working on doing less holding to sleep and nursing to sleep. One tip is to nurse in another brightly lit room, then move to the sleeping room and hold. You can add other supportive methods to help her fall asleep in the crib by first holding her in the crib, then move to a hand on her back. This will be a long process for sure.
6. If it's been more than 1.5 weeks since you think the growth spurt began, you can try decreasing the amount given... so one breast instead of 2 then less and less time on the breast. Especially at the first NW.
Hoping for great, restorative sleep for all in the near future! Hang in there. xo