To pump or not to pump

mjeg30

I have a 4 day old (yah) and my milk came in yesterday evening. They are crazy full and engorged. My baby is feeding a lot but he doesn't seem to be taking enough to relieve the fullness. Is this normal? I am a FTM so this is all new. What I notice is also happening is that he sucks for a little bit, falls asleep and off my nipple. Then I lay him down and a couple minutes later he wants more. Normal? 

 The hospital gave me a manual pump. I wasn't planning on pumping, but should I to relieve the engorgement, or just use the ice to relieve the pressure? The lactation consultant told me to pump when my milk comes in but because I was on Information overload I don't remember why!

 I really don't want to deal with bottles, etc. But want to do what's best.