Falling to sleep while feeding is a completely natural thing for a small baby. In general young babies don't want to sleep anywhere by themselves. You are in the "fourth trimester" e.g. see this information where baby is getting used to being outside of the warm, snuggly womb and they like to be cuddled up to their parents. Sleeping in a big, flat, cold crib is what baby has to get used to. You could try making sure baby is all wrapped up with some sort of sleep-safe swaddle (only until they can roll over), making sure the sleep surface is pre-warmed (we used to put a hot water bottle in and remove it when we put baby down) or babywear for naps so mum can do things. We also researched safer co-sleeping guidance from a UK safe sleep charity as it is much safer than accidentally falling asleep on the sofa when you're trying not to, and allowed us to all get a much better night of sleep.
Over time baby sleep is very much non linear. You'll think you've got it and bam, they won't sleep for a couple of weeks and they'll go back to normal! You'll find yourself regularly wondering whats gone wrong over the next two years.