I wish I had been told-
Get a good lawyer, not a nice lawyer a good one. Tell the GP, HV, School, bank, cc company, services everyone you are now single and get official financial disassociation notice in place (look this up it is fucking important). Get credit reporting notification, it doesn't cost much and you need to know if he is or has fucked up your finances. Apply for all discounts and funding - if you have any friends who have done this family to single parent process get them on board for this bit, they can come over and help with calls and baby and make sure nothing gets missed out.
Change all passwords, add security words to accounts, change security question answers. Assume a total breach and act accordingly. Banish the words "he wouldn't" from your life, he would and has.
This will help it all sink in and get your life back on your terms. Even if sometime in the future you both resolve this, for now you need to put your dc first and get your house in order. He has gone, assume it is for the best and start as you mean to go on.
Change the locks and pack his shit up and put it out. The lock change ban is about denying him access to his property eg if both of you are on the house deeds then legally both of you have a right to access, and also a right to change whatever SO you can change the locks you just can't do anything to legally stop him breaking in. Make it difficult for him. Change the locks, and get one put on any rooms in the house you want private, like your bedroom. Take charge of your house and space, you are allowed to and it will be good for you. Get anything valuable or important to you out of the house.
Feeding a baby when you are stressed is really tricky, eating regularly is a big help. Can you enlist your family/friends to take turns staying with you to help out and to feed you good regular food? It makes life a lot easier for you and the dc and having a third party in the house whenever ex shows up will be a big help in stopping him showing up. You are in charge whilst baby is bf, this is your job and not negotiable - you don't need permission to do what's best for baby or for your children ever. Do your job and don't be guilted out of it.
Prepare for this to get nasty as soon as he realises things are no longer in his control. He is assuming you are at his beck and call, waiting for him and doing the right thing like you always do. There will be some temper and dick swinging once he realises that ain't happening. Stop playing by the rules.
Get a new sound board that isn't him, he is no longer on your team. You deserve better. Do a list of the shit things he has said and done on 1 bit of paper and a list of practical to do stuff next to a list of everything you want out of life (no matter how wild) on another bit. Pretty soon you will only need the second bit of paper, it gets easier.
Tell people you are now a single parent because your husband has left you. The shame is his, don't hide what he has done. You will get support and the more you say it the more you will settle into your new reality.
I would send a very short clear text or email to in laws stating - "As you know husband has left and I am now a single parent. Obviously I hope that this won't effect the children's relationship with you. Please feel free to contact me directly to arrange visits, thanks". Then meet them in public, not your home, so you can leave if they are dicks. And have someone stay at your home while you do because if he is going to break into the house it will be when you are meeting his parents.
He isn't who you thought he was. What would you say to your friend or sister if their husband did this? Would you want them to get back together with a selfish lying shit head who bails whenever he feels like it?
Be prepared to hear a lot of shit about yourself in the next while. Just remember he left and he left the children with you. Family abandonment is judged (not harshly enough if men do it imo) and he will need some excuses to stop feeling the judgement. Some fucked up backwards rationalisation process goes on in these tiny men's minds. It is irrelevant.
What good parent leaves children with a controlling abusive whatever? Answer NONE, if he thinks you are good enough to take care of the children (vulnerable most precious innocent little humans) and be full time carer then that immediately cancels out any shit he says about you.
Look up MrsDeVere if you want a masterclass in dealing with stbxh.