Hi, I could have written this myself a few months ago! My daughter is now just under 4 months and I would honestly say it just takes time...
I was tearing my hair out at first because I couldn't put her down without her crying and I tried everything but sometimes it just takes time for them to adjust to their new living environment as you have to remember where they've just spent the last 9 months! Someone told me to read up on the '4th trimester' and it was really interesting, it helped me understand her a little more about why she was so clingy or upset etc at times, everything has changed for them in an instant and it takes a while to get used to it...it's definitely worth googling!
My best advice would be feed her, rock/sway her, make sure the environment is right for sleep (like previous poster - heat pad, tshirt with your smell, white noise, dark room, not too cold/warm etc) and then lay her down. My biggest mistake was not introducing a dummy sooner, she's only just started taking it in the last few weeks after much perseverance so if you can try now I would definitely keep at it, it's really been a lifesaver at times and I cringe at myself as I originally turned my nose up at a dummy but it helps my little one self soothe and fall asleep (bear in mind babies can't really self soothe until about 3 months old though) and I can't believe I hadn't tried with her sooner as it would have probably got me through the first few difficult months.
We had to keep DD upright for half an hour after feeds because of reflux and I used to sit up in bed with her asleep on my chest after feeds and then slowly put her into the bedside cot and normally she would go down okay, id maybe keep my hand on her chest for a minute or so to let her adjust. I think there's a lot of opinions on co sleeping but as long as you're following 'guidelines' on it and it works for you 😊
My best advice is just try ride it out, I found the first two months really hard in terms of getting her to sleep, finding a 'routine' with everything (including myself) and learning what this little one wants. I promise it'll get better but the important thing is to stay calm as they can sense your mood and it's not worth getting upset over as they will eventually sleep and one day you'll find yourself in a good wee routine, it happened naturally with us and she is now sleeping 8pm-11pm, dream feed then 11.30pm-6am xx