I think by 6 weeks everything is pretty settled, and then they start their growth spurts and they need more, so it gets crazy again. I don't really know though, my story is different from most.
I had the same issue with you, and couldn't take it. I started formula feeding at that time, and then a month later decided it was an incredibly dumb decision to stop and I started the process of relactating (with the help of a midwife that has a lot of experience with breastfeeding mothers). This is much more difficult than just working through breastfeeding issues, but I think I have some tips to help you keep at it. What I really needed was a break from the constant breastfeeding. I couldn't pump (my body had a really hard time with it and just wouldn't produce it for the pump), but if you can, you should try to get someone else to feed your child for at least one time during the night, maybe another time during the day as well. That way you can get a full 2 or 3 hours of sleep and not be nursing (or more if you are getting at least that). You could also try doing this extra feeding with formula if you can't pump. Just one small feeding a day won't hurt anything, it will let your body and mind recover a little, so that you can continue until you get through this part. You could have a really slow flow, which is why your baby is falling asleep (I think this was my issue). Once they get older this isn't as big of a deal, but for an infant, they fall asleep. See if you can hold out a little longer when your child seems like he is hungry, try to get to an hour between real feedings (where he doesn't fall asleep), and then you can start stretching out the time a little more. It is really important to just make it through the next couple of weeks, and then it should just get better, so just do what you have to, and get through this hard time. Good luck.
Oh, one more thing, cosleeping for naps really helped me as well. That way you both get to rest, when he wakes up, you just pop him on, and it isn't really taking you any extra effort, so you can get more rest (lack of sleep with constant nursing will destroy your sanity and make you think you can't continue if you already have doubts and problems).