OPM, I also have been having issues with breastfeeding. My son is very sleepy and falls asleep probably about 80% of the time I try to nurse him, sometimes after sucking for only a few seconds. There have been a few times when I've gotten him to nurse for 40 minutes, too though.
He was born 2 weeks early and I have always thought my edd was a little too early and should be later so probably really at like 37.5 weeks. And he was pretty tiny. The nurses said this behavior is typical of a somewhat early baby.
I have been working with a lactation consultant and what they are having me do right now is first try to nurse him. Then if he falls asleep or won't nurse, immediately give him a bottle of pumped breastmilk (or formula as a last resort) and pump. Also, get as much skin to skin contact with the baby as you can. (I know it's not always possible but even getting yourself down to a tank top and him down to his diaper and holding him helps). Since he was early it took a little longer for my supply to kick in so I've been dealing with that issue as well.
I have read about the nipple confusion as well, but it doesn't seem to be bothering my baby. In fact, when I let him suck on a pacifier for a few minutes before I nurse him, I find that he actually does better. And even though I have introduced some formula, he still prefers the breastmilk to the formula. (I dont blame him, the formula smells pretty rancid while the BM doesn't seem to smell at all).
In the end, making sure he got the calories he needed has only made him stronger and he is actually getting better at breastfeeding, not worse. Don't let all the stuff on the internet scare you. Yes, you want to BF but you also want to make sure your daughter gets some food in her in the meantime.
My son lost too much weight and that is why I ended up having to supplement. And he was a little peanut to begin with so I was really worried about him! It wasn't my first choice, but not every baby "gets it" right away and I will do what I have to do for him to eat!
Anyway, I'm only like a week into it so I'm no expert but I thought it might help for me to share my story. I'll keep ya posted on how it all works out! We are meeting with the LC again today to check his progress/weigh him. Wish me luck!
Oh yeah, also I ended up renting a hospital grade pump. It works really well and I had tried pumping with the manual one and got nothing! It cost me $75 per month but I figure it is worth it if I can get him to BF. Hopefully I"ll only need it for the month! She said if I had been on WIC or had different insurance it would have been free. Mine didn't cover it but eh oh well.