You are not a crap mum!!! There, just to get that one dealt with first...
You poor thing it must be so tough, dealing with a newborn and an unsettled toddler on your own in this baking heat (((hug))).
It sounds like your baby at least is just being a baby, feeding well and wanting to be held a lot which is normal and as it should be. Not convenient though!
Sounds like your boy is struggling a bit with the new arrival - also completely normal and nothing you have or haven't done. He's been the centre of your world, and now he has to cede the position - he isn't going to like it and he is going to kick back! It shows he is used to and feels entitled to a lot of attention and focus from you which is actually a massive endorsement of your mothering xx
If you feel up to it, do go to the park - being out in the air will probably help the baby sleep, and then you can focus on your son and play with him. If he does go a bit Jaws on the other kids, their mums will understand - two year olds do that sometimes and we've all been there when our children's behaviour is less than ideal. Staying cooped up will only make him more antsy and you feel more trapped, so even if it's just a half an hour stroll round the block do try to get out once a day if you can.
Something else I read is that for breastfeeding mums, it's a good idea to set up a bag/box full of 'special' toys or even treats that only comes out when baby is feeding, on the proviso he sits nicely with you and plays gently while you are feeding. Is that something that might work for your son?
These are all just ideas off the top of my head but main thing to remember is you are fine, it's a big job and you will be feeling tired and overwhelmed as it is. Be gentle to yourself and be kind to yourself. I find it helps to imagine myself as a dear friend telling me exactly what you've written - you wouldn't tell her she was rubbish, you'd give her a hug, make her a cuppa and remind her how well she was doing. Be your own friend! xx