Congratulations on your DS! 3 weeks is such a difficulty age, and I am so sorry you don't have more support. DH and I currently live in the UK, but we are foreigners, our families live abroad, so except for 2 weeks when my mum came to help us, we've been pretty much on our own. DS is now 1.
It really does get better, as someone else said around 6 weeks, then 12 weeks, then 6 months. I know this sounds like ages away, but it goes very quickly. I found the easiest way for me to deal with it was to just accept that things will be difficult for a while and I will get almost no sleep, and just work around this really. I co-slept and let DS nap on me for the first 4-5 months or so, and went to bed really early. They change very quickly.
I also found the mother and babies groups a lifeline. I know I didn't have a lot in common with most mums there, and it was only small talk, but as I said our families live abroad and I would have been very isolated otherwise. Are there any such groups near you?
Also, I think your DH's temper is worrying, he really needs to pull his weight and help you at night, even if it's just for a couple of nights a week. If you manage a few hours of uninterrupted sleep, you will feel much better able to cope. You have many unsettled nights to come, colds and teething and developmental milestones, and although it gets much easier, you will still be sleep deprived. Also your DS is likely to get more challenging as he gets older, and although the crying over nothing will stop (mostly!), he will have temper tantrums and will be very difficult at times, so you will need your DH to be calm and support you.
Sending you an un-MN hug!