can'tthinkofagoodname, your situation does sound similar to my DS, but I can't guarantee that what worked for him would work for you. This is a bit of an essay, but I hope it's useful.
DS was (in hindsight) pretty colicky ie he needed soothing for more than 3 hours a day on average to prevent crying. Nights weren't so bad, but in the day he'd shriek his head off before dropping off to sleep. Discovering the sling at 9 weeks was one of the biggest reliefs of my life, as it (usually) worked and without so many tears.
As the colickyness gradually improved, he started to feed to sleep at bedtime (but not in the middle of the night, and never during the day) - somewhere between 10 and 12 weeks, IIRC. Then it improved some more - he no longer shrieked in the sling for the early evening nap he needed before bedtime. (He still needed the nap, but didn't really yell any more.)
Then we were advised by a paediatrician friend that if we were having to fight him to sleep, we were probably mis-reading his tired signals during the day which was why only the sling worked. It turned out that we had been trying to get him off to sleep before he was really tired, which was why other methods failed to work. Only the sling would induce drowsiness in a baby who wasn't really sleepy.
We had been trying to get him to sleep when he started yawning and grizzling about 45 mins after he woke up (which we assumed meant he was tired). However, when we tried keeping him awake for longer (about an hour and a half) it turned out that it was the second lot of yawns / grizzling that meant he was tired. If we put him in the sling then, he would drop off instantly rather than it taking a while.
Once we'd got a better idea of when he was tired, I tried getting him off without the sling. Various methods then worked:
- car plus dummy (someone had to hold it in his mouth as he couldn't do that himself) with a muslin over the front of the car seat;
- pram on very bumpy ground plus dummy and a muslin over it.
- rocking - I'd swaddle him, lie him on his side across my lap as if to bf him, with his face into my elbow, put in dummy (and hold it in with my elbow), put a muslin over my shoulder/his head to block out visual stimulation, play white noise, rock madly and pat him.
All of those methods had previously failed repeatedly, which I assume was because he was still at the colicky stage and horribly overtired. However once he got the hang of going to sleep fairly easily (in the sling) it was relatively easy to help him get to sleep by other methods. There was some crying involved but it was protest crying rather than hysterics, IYSWIM, and it worked. As he got used to it, the crying got less.
I then tried feeding him at the point at which he was evidently tired, and he happily fed himself to sleep on my lap. He had always refused before - because he wasn't actually that tired, and it wasn't actually that long since his previous nap. (Both then and now, if he's overtired, he just refuses to feed, or feeds for a short while and then yells - he won't feed to sleep.)
I had a stage of feeding him to sleep for every nap and progressed to being able to put him down before he was fast asleep. Then I managed to rock him to sleep swaddled & with a dummy, but without the muslin / white noise / patting. As he got more used to it I was able to ditch the dummy. Gradually I was able to put him down sooner and sooner, so recently I was able to rock him until a bit drowsy (when his eyes start to roll) and then put him down and he'd do the rest on his own (with a bit of shush patting if necessary).
If your DD likes the sling, she may well like being swaddled, as I think the sensation of being wrapped up tight is similar - certainly my DS loves both.
I don't know if any of that will help, but there's more on the daytime sleeping habits of colicky babies here. It takes some babies 4 months or so to get over the unsettled colicky stage.
I would have a go at trying for a nap after around 1.5 hours awake (maybe only 1 hour after waking in the morning). Babies find it easiest to get to sleep when they are well-rested, so they ought to find the first nap of the day easiest.
HTH!