After being a long time lurker on this forum , I felt compelled to leave a comment as I found myself in a very similar situation.
Background : Was burnt out having spent ten years working in a FAANG company, father was undergoing treatment for cancer and my wife was due with twin boys!
Couple of thoughts:
Talk to friends: Two months before the boys were due - I sent a text to three of closest friends to say that I was feeling overwhelmed by life and needed to chat to someone. This was by far the best thing I did . They had great advice but more importantly I was able to get my concerns out in the air and examine them in the cold light of day. If you are not comfortable talking to a friend , reach out to a therapist or even a helpline. It really is good to talk.
Take to mgmt in another department or in your industry: Taking a step back and thinking about your scenario from the company's perspective. Talk to managers in other departments or outside your company to discuss how best to manage this situation. They will give you a steer on how best to navigate this scenario for your mutual benefit. For example if you know you are expecting a child - have a plan in place for how you will ensure your work can be completed in your absence , who could take over etc. Have a plan and communicate that plan on a regular basis.
Priorities your family / depriortise work : For me I made the decision to move horizontal and to put off making a significant career move for a year. Making the decision not to make a change freed me from the background stress of feeling I was stuck. If you can - take what ever time is available for you to be with your family.
Cut yourself some slack - Before the boys arrived I trained everyday, went to the theatre , did self study and had a number of commitments outside of the house. I decided to depriortise a number of things and dial down the pressure I was putting on myself.
Surf the wave: By nature I am a problem solver but something's can't be solved in the short term or require trade offs in the long term. During the most stressful periods I visualised myself surfing the wave ! If I tried to fight it , I would be crushed - but by going with the flow and enjoying the ride - it helped me cope.
Finally - here is the secret no one tells you. We are all laughably under prepared ! I was worried about not being the type of father I had but what you realises is Parenthood is a journey not a destination . You learn on the job.