My baby won't let me put her/him down
I hear this a lot, first of all, your baby is trying to tell you something. Key into your baby, is the baby still hungry, have gas, need a diaper change, have reflux or just needs consoling. Just go down the list until you figure it out.
If you just fed the baby, then offer more milk, if they turn away or suck and then turn away it's not hunger. Infants self soothe with sucking, so if they are not actively feeding when you offer then it is not hunger.
Check the diaper one more time, often babies wait for a brand new diaper and then go. So you might have just changed it but check again.
Also note how your baby is acting, squirming, arching back and kicking legs up are usually gas pains. Try burping your baby one more time, a NICU trick is to the lay the baby down for about minute, massage the tummy gently and then sit them up to burp. This sometimes helps to bring the air bubble up. If they are still gassy, then a warm small hand towel on their tummy helps the gas pass faster.
If you suspect your baby has reflux, holding upright for 20-30 min after a feed or putting them in an inclined position such as a swing or bouncer would help. Be mindful not to turn it on for at least 30 min as sometimes the motion will make them throw up. This will give the baby time to start to digest the milk.
Once you have gone through the process, your baby might just want to be held. Some mums love to carry their baby all the time, if you do a fabric baby/mum body wrap is a great idea to keep your hands free and it's easier on your back.
Babies have different cries for different needs and as you become more used to your baby, you will know what the baby is telling you. All this takes time.