I agree with PP. DD was this way at that point, and kinda still is. I had/have such a hard time keeping her awake and did the cold washcloth and tickling things. Also, I find that rubbing her back vigorously works a little. I don't mind her eyes being closed, but as long as she's sucking, she's awake to me. And mine does gulp a lot at the beginning of feedings, so I know she's getting a ton of milk. (And as long as your LO is growing, I wouldn't worry so much. DD was the same way, and I was so worried about her growth, but then I found she gained three pounds in the month since her birth!)
ETA: If you're really that concerned, call your pedi and ask to bring your LO in for a weight check for reassurance.