I felt like this, OP. I had stopped antidepressants during pregnancy and struggled on for a while. I fantasized about being in hospital for weeks on end, or about one or more children somehow just disappearing as if they’d never been there.
Your baby is a month old so even if you’ve upped your ADs since birth they might not have kicked in yet.
I second PP’s advice to get out just a little bit each day. One way I made this happen was by deciding to go at a particular time. If I hadn’t managed to get them dressed by then, I went anyway. As long as I was dressed-ish and everyone had a clean dry bum off we went. Sometimes to the park or just for a walk around the block. I gave up on taking changing bags and snacks and life was way easier. (Obviously I took a changing bag if going out for a long time, but not just to the v. close park.)
Definitely tell someone professional, but fingers crossed the ADs will kick in and help you fake it until you make it.
When the baby naps you probably want to instantly devote your attention to the toddler or the endless to-do list but try napping too, even if it means lying beside your DS as he plays and just dozing.
I noticed you said the baby isn’t even that difficult in terms of newborns, but they are still massive shocks to the system.
Be gentle with yourself. It is normal (even if it’s an “abnormal” normal that lasts longer or is more severe than it is for most), but normal doesn’t mean it will necessarily go away by itself. Babies are fucking hard. Talk to your GP and if they say it will get better by itself talk to someone else until you get help. Get as much help as you can, drop what you can. Can you invite people round? Just having other adults there helped me freak out less and sometimes they even folded a basket of washing! Plus they can hold the baby and let you cuddle DS.
Keep posting, OP. 