I had so many anxiety attacks over sleep when our little one was first born. I am here and ALIVE. We survived!!!! He's about to turn one this week.
You will find a way to cope, and you will get your own system up for when you and dad sleeps once you're home and in your bubble.
For us, I went to bed with baby at 6pm every day. I would co-sleep with him, and dad would help with nappy changes and rocking to sleep if he didn't want to feed. I didn't go on my phone, I avoided sugary snacks and caffeine as I had major issues sleeping, even when baby was sleeping, due to the PND anxiety. Between 6pm and the next morning I would get enough sleep, although completely broken, to get me through the day. There were nights I didn't sleep at all in the beginning due to the PND/A. But take each day at a time.
We EBF. It is incredibly tough in those first weeks, but for me has been worth it in the long run as life is so easy when you don't have the faff or cost of formula. He still feeds on demand now, and I plan on feeding as long as it continues to work for us. It is such a good tool when they are sad and need comfort. I hope to keep going until he's at least three, as the comfort he gets from feeding is like magic and supposed to really help you get through those "big emotion" toddler years.
Fussing at the boob, crying and wanting to feed what feels like 24/7 is all totally normal. Many people give up when the baby is unsettled on the boob and think there is a problem, but that is usually the baby working to increase your supply to meet their needs. You just have to give in to it; sleep whilst they are suckling if you can. Either side sleeping/feeding or even chest sleeping/feeding (we did this a lot in the evenings (6-10) and baby's dad would watch TV next to us and keep an eye on us.