It is totally normal, you're right And yes, he will grow out of it pretty quickly.
He is a tiny baby who was tucked up tight inside you juat a few days ago, warm, cosy, comfortable, food on tap, comfort on tap. Now he's suddenly in this huge bright open space with all sorts of strange noises and sensations around him. It's no wonder he just wants to be close to you, really.
Make the most of it, I remember how frustrating it is when you spend all day with a baby snuggled on your chest but the newborn bit flies by and it will be over all too soon. I miss those lovely snuggly times.
The sling is a good idea, also swaddling tightly can ease the transition from arms to bed. I used to put a blanket on my chest and lift baby and blanket into the moses basket so there wasn't a sudden change in temp from my warm chest to a cold sheet. A t'shirt that smells of you wrapped around the mattress also helps.
The constant holding eases off by about 6 weeks iirc and they are happier to be put down for short periods.
Congratulations on your baby!