You're not even slightly rubbish. Not a bit. If you were rubbish, you'd let your baby cry, roll over and go back to sleep, tell him to shut the fuck up, hit him, neglect him. You're doing none of these things.
You do sound very down, and the rejecting help because you feel you 'should' be able to do it rings huge PND bells for me (that was me, when DD was little, it was HELL).
Listen, you aren't magical. You don't get born knowing how to soothe a baby. It isn't conferred upon you at birth. Some babies are content, some are clingy, some are screamy. I had a screamy one. Bad luck me, but not my fault.
Practical things: get HV to check there isn't tongue tie or reflux. Call today and get them to come to you, if possible. Find a breastfeeding support group - often they are hotter on tongue tie/latch problems/reflux, and hey, more help is better than less, right? I went to a group, sobbed uncontrollably on an amazing lady who took howling DD, and cuddled me, saying 'you are doing it just fine my love'. DD didn't have any 'issues', she's just a screamer, but that support meant so much to me.
See your GP and talk about how you're feeling. Be honest. If it is PND, there are loads of things you can access to help.
Sleep wise, it changes so fast when they're little - it might resolve quickly, it might not. Do you have a bouncy chair? Sleepyhead? If you can throw money at it, you could try swaddling type aids, chairs to hold him more upright, different slings. For DD, tucking a v-pillow around her (budget sleepyhead!) worked, but it was trial and error. Lots of error.
Non-practical things: have a hug. You're doing fine. You will get through this, and your DH can carry more of the burden for now. He's proving that by doing it - let him show his love like this. Ask for help. If you do go out, and get overwhelmed, pick another mum, preferably a smiley approachable looking one and say 'I'm about to burst into tears, could you possibly hold him for a sec, let me regain my composure.' I guarantee 95% of mothers will get it instantly, and lunge for DS to give you five mins peace - I know I do. Don't feel ashamed and hide it, don't feel like they all 'know' what they're doing - they don't. Talk about it. Practice saying 'I need a break for a bit', everyone does. I was at a soft play the other day, mum next to me had a three year old who palled up with my DD. She also had a newborn, and was trying to feed him, getting all hot and red and weepy. I said 'god, I remember those days - and not fondly. Can I do anything?' I held her DS while she went for a wee, grabbed a cup of tea, then came back to play with her DD for ten minutes, and yes, he screamed for me too, but it was fine, and he was fine, and she was so much calmer and ready to wade back in. People have done the same for me, they want to help. Don't PLEASE don't struggle on when you can let it out - just remember, most parents have had at least one moment (for us, six months) where they're at a loss to know what the fuck to do, where they feel inadequate and lost.
And, to cheer you up, once DD could move - sit, roll, crawl, she was honestly like a different child. She slept, she smiled, she laughed. At nine weeks, I thought that would never ever happen. It did. It will for you too.
Sorry, massive essay, you just struck a chord for me. So much love and luck to you.