Certainly worth trying a swaddle if you want to. FWIW for the "extra layer" I always used a short sleeved vest under their other clothing, I didn't see the blanket as the layer if that makes sense. I think the ready made ones are supposed to be easier but there are loads of videos of how to do it with a muslin or thin blanket on Youtube as well :)
WRT overheating the best tip I ever got was to feel their chest or the back of their neck so poke your finger down into their clothing. If they feel cool then they are too cold, if they feel sweaty then they are too hot, but they can feel quite warm/hot without it necessarily meaning they are overheating. They won't be in immediate danger from overheating (nor being too cold) so there is no harm in you getting it a bit wrong - try it out and see how it goes, you can always just take a layer off if they get too hot.
Don't worry as the epic cluster feeding won't last forever, it will stop by itself, although it may come back sporadically. It's horrible when you're stuck in it. Maybe DH can take him out for a walk, or even a drive for an hour or so, so you can get some sleep if it happens again. At least if he cries, you won't be able to hear and movement often settles a lot of babies. Get tanked up on sleep in the day as well either when the baby sleeps or when someone else can take them for a bit. In this newborn period I find it's best to ignore the clock and go by body needs so just shower/sleep/eat when you get the chance and fill in the baby's needs around that!
Another option to look into which works for some babies is side-lying breastfeeding. That worked for me in the early days especially with DS1. It just means that you can snooze a bit while they stay latched on. It's not always as restful as a proper sleep but it's better than nothing. Just don't swaddle while you do this, the baby needs to be able to move naturally.