Honestly OP I have been exactly where you are, except I breastfed and baby would only eventually settle for me. Naps were non existent, night time sleep equally horrendous with either frequent wakes or split nights, or both!! It was this way what felt like forever! I had post natal depression and was absolutely spiralling.
We paid for a sleep consultant and her sleep did temporarily improve to three wakes a night, but then you have teething, sickness, regressions and back to square one and I couldn’t go through with it all again. So I waited, and responded, waited and responded. First her day nap improved when she dropped to one nap at around 12 months old. She would start having a longer nap of just over an hour. Then suddenly, one night, out of the blue, probably around 19/20 months, she just slept through. I didn’t do anything differently. I just kept responding to her.
You are letting her sleep control your whole existence, as did I. It’s so so so difficult but if she is fighting sleep so hard, don’t fight with her. From my own experience I personally don’t believe in overtiredness anymore thanks to my sleep gremlin. I honestly think it’s a term banded around to make desperate parents part with their hard earned money.
What I do find strange though is that your baby wants to be carried all day but hates the sling/carrier. Have you been to a sling library or tried a back carry? Coz a sling is normally the perfect solution for the clingy baby.
I know you really don’t want to hear this, coz I didn’t want to hear it either, but all my girl needed was a secure attachment and time. Try to not let her sleep control your day. If she is fussing, let her fuss. She is a baby and that’s what fussy babies do. If you can afford it maybe look at a nursery or childcare one day a week so you can have some time to refresh yourself each week. If you haven’t already maybe try a back carry. If she is happy but awake in the pram, leave her in the pram awake and enjoy the park with your son. And be kind to yourself. You are in the thick of it now. It’s hard! X