You are not rubbish!
You're only job is to care for your son, the rest of the house, all the cooing etc are not your responsibility yet. So forget it, you are not failing!
And when I say it's your responsibility to care for your DS I don't mean that it's your job to get him to sleep, it's your job to be present, and to care for him, and you're doing that!
By the sounds of it you need to have a few days recharging your own batteries and then try a new strategy with your ds,
Over the next weekend tell your dp that he is to take your ds out all day, bringing him home only for feeding.
He can walk around the park. He can walk around the block. Repeatedly. But he is to walk that baby, he can go to a coffee shop. He can read the paper on a bench. It doesn't matter what he does, so long as it's not super stimulating for ds, and it's outside of the house, so you can have a bath and sleep or be peaceful on your own For all the time you don't need the baby latched on.
You'll feel better after two days of this.
My husband had to do this every weekend for the first while, and ours was a January baby, your dp can do the same without being buffetted by artic winds.
Then think about the following:
sleep cycle is forty minutes, if the conditions he experiences then are different to those when he's fallen asleep he'll wake.
Our DS was a difficult sleeper, we did cosleep, he did latch on frequently but I got to sleep. But that doesn't sound like it works for you, and tbh I'm not sure I'd do the same again.
But, what helped get his nights better was getting his day sleeps better.
3 hours awake 2 hours asleep 3hours awake 2 hours asleep etc.
To begin with, when I tried to fix our no sleeping ds, it seemed like I spent all day trying to get him to sleep or facilitating his sleep.
For me that meant figuring out how I wanted him to sleep and then paying lots of attention to making sure the conditions at the 35-50mintue mark were the same as when he fell asleep.
For us I found the pram was practical, I like walking.
I'd feed him up, and then go for a walk. I used to tell him that if he wanted to go to sleep this was a good time, that I'd keep him safe and that he could relax. I then would rhythmically say things like 'relax your arms, stretch out your legs, relax your fingers, and your toes' in a calm, Rhythmic not stimulating voice. At some point he'd fall asleep. To begin with I made sure that the pram was still rolling and not in a noisy place at the 35-50min mark, even if I stopped once he was in the next sleep cycle. .
He began to fall asleep quicker, now he often falls asleep within two minutes of me suggesting it's a safe time to sleep (but I now add 'when you wake up we'll be somewhere/ it will be time for some fun)
At night I tapped him on the belly as he was falling asleep, and again at the 35/45 min mark (during the shift in sleep cycles time.) after a while this wasn't necessary, and the night sleeps got longer.
I think Joanna blythman's book might have been where I got this approach from.
It certainly calmed me down and helped me understand more about sleep.
You are doing great.