He's only drinking enough to satisfy the hunger pain, and not actually get full if he's falling asleep while nursing, so it only lasts him about an hour before he's starving again and screaming.
Do everything you can to keep him awake while nursing, turn on lights, put a cold washcloth on his cheek, sing some happy, upbeat songs, etc. You can't let him doze off while eating. Even try a different position to lay him in, if he snuggles up all comfy next to mom, he's going to doze back off. Try something like holding him more upright rather than in a laying position, or even sit him up in moms lap facing the breast to feed.