Hey there fellow human.
Here's my little bit of help I can offer. I haven't had a kid, yet, but I know what you are going through. I've been lucky enough to have been able to fall back on things like employment insurance here in Canada to be able to take some time to myself to let my inner angst out and deal with it constructively. You probably don't have that available, so here's my advice for in the meantime.
1. Find at least 3 hours in a day that you can dedicate to yourself, nothing else. This will not be easy since you have a kid now, but I believe everyone needs to have at least 10% of the day dedicated to themselves, otherwise burn out is very possible for anyone.
2. Ignore people saying this is 'just how life is'. It's not. There are plenty of people out there living perfectly content lives still having to deal with all the same things you are, and possibly more. My parents had to drop everything and move us kids (4 of us til 5th was born) out to the bush just to make ends meet once upon a time ago. There were hard days, and plenty of tough times, but I never heard my mother or father complain about burn out or anything like that. Not because they were tough as nails or anything like that, but because they were able to do what they needed to without interference from every joe dick jane and jerry. So my advice here is to just do what YOU think you need to do, provided its not going to harm anyone else or yourself. That's liberty. That's freedom. You're burning out because you are lacking these things (IMHO).
3. Find something you truly enjoy doing that isn't just consuming more content. The internet can sometimes lead to worse burn out for some, due to the nature of how we use it. If you must be on the internet for whatever it is you enjoy doing, just at least avoid social media. That will drain a person faster than a lot of anything else. (Again IMHO)
4. You don't need therapy via some specialist. You just need someone who will listen when you need to vent. Just remember to be there for that person too when its their turn to unleash their troubles. That's what we're here for, for each other.
5. In regards to learning something new. In the words of Shia Labeouf "Just Do IT". I know its hard, I struggle with it too. I have 3 books on different aspects of computer sciences just waiting to be used. Haven't touched them. Can't be damned to. Not because of burn out, but because there is always something else that I want to do instead. So I really should take my own advice here.
6. In tandem with avoiding social media... do the same with the news. We need to stay informed, but the informers aren't always really doing what they do to inform us, but to keep us scared of what we can do nothing about except worry.
So just do whatever it is you want/need to do instead of wasting time trying to stay informed. A large degree of it is pure unadulterated bullshit anyways.
7. Listen to Donkey Kong Music. Or whatever music you find enjoyable and relaxing. I find Donkey Kong tunes to be very much this.
8. Again, ignore the people who say "this is just how life is". It's not. It's just how life is right now because people have consigned themselves to it. Yes, there are plenty of things that can make life terrible, but that doesn't mean we need to dwell on them, or even accept them. Our greatest achievement as a species on this planet has been our ability to transcend these issues; or at the very least withstand them until we can transcend them.
9. I leave this here for future edits if I decide to add more.