Erm when I did cc it was 2 bad nights, with quite a bit of crying, a few nights with a few grizzles, but no outright crying and then 9.5 years of excellent sleep. He wakes up , sticks in thumb, sucks and falls back to sleep, he is 10 now and still doing it. He changed from a hyper grumpy baby who wouldn't be put down or nap in the day, into a calm happy smiling baby in 3 or 4 days, and he had good day time naps too. Even his feeding improved!
I never have had to do it again and he's slept well through teething, illness, etc etc. it did teach him to go to sleep by himself. And he's still a cuddly affectionate boy, likes to snuggle up for a while before bed. Luckily dd was always a good sleeper and I didn't have to resort to cc for her.
I hated doing it and I can remember Dh and I both crying in bed listening to DS crying, but it worked and I think it was the right thing to do.
It sounds like you need to try this from the point of view of your health, you can't be a great mum if you're ill.
Would you consider ff? That would allow your partner to feed while you rest up?
And I agree with the posters who say good day time sleep is related to good night time sleep. Solve one and it's quite likely the other will improve.
Good luck, cc could help you and your LO.