I have an 10 week old who is like this, she's actually got worse: she would sleep lying down at the start but has developed quite bad silent reflux.
You're still recovering from delivering her, you need rest and support.
Things that helped:
At first at least, don't try to solve the problem of the baby not sleeping, try to solve the problem of you being exhausted. So: I'll try and start the cluster feeding early while the toddler is given dinner and bedtime then DH will hold the baby upright (she'll sleep in that position if held) from about 8pm till midnight/1am, then try to put her down. If she automatically wakes, then at least I'll have had at a minimum 3 hours. He'll then try and take her down first thing. We've also found (he's WFH at the minute) that she'll often sleep for a couple of hours on him in the sling in the day, meaning I can nap. When they're so v little, they're often happy to be held by anyone, so draft in anyone you can to come by for tea, cuddle the baby and let you nap. Even letting you scroll mindlessly on your phone can be restorative!
In terms of what is causing it, at this stage, I think its probably just a new baby thing. We got a white noise machine which helped a bit, and those gro snug zip in swaddles which also helped at the start, as did dummies. I've also found one of us keeping a hand on her tummy while she's in the next to me can help.
As time goes on, if you think its digestive discomfort, I've had some success with baby Gaia probiotics. And also, breastfeeding was great for me in figuring out allergies: my eldest had them so it was easy to spot by doing an exclusion diet. My next step is baby gaviscon, then back to the GP for medication, but we're at 10 weeks now.
I was a zombie with my first baby, who was the same, and I also couldn't handle co-sleeping. I resort to it at about 5am on this baby if Im in pieces, as I think I can do it safely but I'm v wary. However I feel much more human because I know the minimum I need to get through it and I know that it ends eventually. Good luck, you'll get there, don't think about things you 'should' be doing or ways to fix it, just focus on getting your own needs met and doing whatever you can with the baby to survive the first few weeks and I promise it will get better.