LO sucking not swallowing, help!

Alesia
I'm at a total loss here... I've been trying to bf my little girl for almost a month now and after many visits to a LC, pediatrician, specialists I don't know what else to do. I'm convinced it's because I'm stupid or broken. I'm feeling very dicouraged at this point. I've had many look at my latch and they say it looks fine, and of course it doesn't hurt when she latches on anymore, but I really do not feel like she is taking in enough breast to get milk. Because like I said she is suckling, but not swallowing, and then of course falling asleep at the breast. But, when I take my breast out of her mouth and check my flow I have my let down and I know I still have PLENTY of milk left for her. Especially since I get at least 3-4 oz every time I pump. I've already had a LC look at my letdown as well and they said everything is fine... so what gives? I don't know what to do. I'm tired, I'm sick of pumping every single time I feed her. I just want to bond with my baby... is this too much to ask?