((((hugs)))) honey, I know it really is hard. It's not that he doesn't want to sleep - he doesn't know how to! I know it sounds crazy, but tiny babies really need to be taught how, and the more overtired they are the harder it is for them. You are doing great work to teach him.
At six weeks I would try to make sure that he is never awake for more about 1hr-1hr10mins max. So that means you need to be wrapping him up in his swaddle and starting to shush-pat when he has been awake only about 50 mins. If he had a short nap, that will need to be even shorter awake time, so literally just feed, change, and get ready to sleep.
Re the sling, it's a really good idea at this age for him to catch up on sleep. I would try to get him to sleep with shush-pat for at least one nap per day, but if it's a big struggle, he could have some of the rest of his naps in the sling. You don't want to come to rely on it, but at this age the most important thing is that he gets enough sleep, it doesn't matter so much how he falls asleep. When he is about 12-13 weeks old you can start again with shush-pat and he should start to find it easier.
I don't know where you are in the world but the Miracle Blanket (US) or Mothercare Swaddle Blanket (UK) is brilliant and keeps them really tightly wrapped. My LO was always huge for his age and very strong and it kept him swaddled til he was 4-5 months. The swaddle needs to be really tight, he shouldn't be able to move his arms at all, like a solid little log of baby.