Does my 3 wk old have a milk allergy or digestive disturbance?

My son will be 3 wks on Monday and though he is my 3rd child i have forgotten some things. I am nursing so i know that runny yellow diapers are normal but what i am wondering is about the number of them. Every diaper that i change, up to 5 during the night time, are poopy ones. Whether it be a full blowout or just a "skid mark" there is a little in each one. I worry because i know he isn't eating too much, since he falls asleep most the time only 6 or 7 minutes after he starts nursing. So is this normal or not? I am taking him for his check-up tomorrow but thought i would ask you all first.