Based on what you describe, I would strongly disagree with your vet on this. It doesn't sound like a behaviour problem at all - it sounds like a fear response (and a perfectly reasonable one at that!).
You said she's growling when people pick her up...
don't let them pick her up! She's telling you loud and clear that she doesn't like it.
One thing you've got to realise about dogs is that they're not people. We primates view hugs, cuddles, picking up babies, etc as affectionate actions. Well they're NOT affectionate actions to a dog - they're a threat. Yes, really
It doesn't mean that dogs can't be
taught that this sort of thing from people is not the threatening gesture it would be if it came from another dog. Of course they can, and most will come to accept (some even enjoy) those sorts of huggs (or anything else involving wrapping your arms around them) from people. But this doesn't happen overnight, and it doesn't happen just naturally.
I would suggest a few things:
1. Don't let the puppy on the couch - not if you've got a baby and are going to end with child and puppy competing for the same space. The pup will be perfectly happy with a bed of her own (which should be a people-free space) on the floor if the rules on who sits where are clear from the outset.
2. Don't let people pick your puppy up - she's clearly not comfortable with it, and being forced will increase the fear and her response to it. If someone picks her up anyway and she growls, MAKE THEM PUT HER DOWN.
3. Start teching the puppy that this sort of thing from people is not threatening. Forget about picking her up for a bit, but instead let her gzet used to being petted on the body. Then the head. Then sitting with an arm over her shoulders. Then move up to a loose hug. And so on. Actually, you may well find that by the time she's comfortable with all that, she's too big for being picked up anyway

It is, however, vitally important that she does learn that this sort of stuff from people is not threatening behaviour, especially with kids in the house. It's not in the least important that you ever get round to picking her up though, so go as slowly as she seems to need/is comfortable with and you'll get there in the end.