If you do take it, please consider doing extra work during good days (if possible). Babies are unpredictable and get sick/ extra needy randomly. Do your best to do extra things when your baby lets you, so you could afford to sleep/ rest / survive when baby needs you.
Your dp needs to do more at home, batch cook now and online shopping is good help. Try doing/ teaching your baby mix feeding. Have a routine for baby if possible. With ds1 who was feed on demand which was very predictable 3hr feed (who took 45min on breast for the first two months). He also had one formula feed at around 9pm by dh which put him to sleep for 4hr. So I slept from almost 7pm till 1am of unbroken sleep. It was great. Until vaccination time, sleep regression at four months hit. But that was nothing.
Ds1 used to entertain himself a lot and loved watching me do things. He was very content being in his bouncy chair. I used to narrate my life to him as he found comfort in hearing my voice. He hardly cried.
Ds3 was a completely different story and had reflux. The only time he wasn't crying was when he had breast in his mouth. He didn't sleep (I Google can you die from lack of sleep). Took him to the gp and the third try of medication helped him. To be honest they all did help but the side effects were counter effective. For example, the first one, stopped him from vomiting but gave him diarrhoea.
Get your dp to save up as many annual leave as possible and book vaccination on fridays. The first day if paracetamol is adviced it might be easy. Paracetamol made my sons sleep for longer.
Try all the safe advices, to get baby asleep. Ds3 feel asleep in his car seat for 3hr after driving until he feel onto deep sleep. But I only realised that when he was four months. Before that it was feeding every hour and he only accepted breastfeeding no bottle. We co slept as I needed to save the little energy I had. Ask dp to take baby out of the house if baby is a screamer, as it can be very tough to nap/ shower/ rest while hearing baby crying for you. I was lucky as I had a good neighbour helping me dropping & picking up ds1 from nursery (I have repaid the favour but it was nice having a child at same nursery as a neighbour). I also had big support from dbro (who I added to the car insurance so he could drive ds3 until he feel asleep) and dsis when they were off from work/uni. Accept help whenever, possible and feasible.
If you end up with a c-section walk as soon as you are able. They say to quicker you are mobile the quicker you heal. Well that is what my midwife told me.