Do you have a room you can shut off, so your toddler can't wake the baby if he sleeps. Get the milk down him the best you can, bring pushchair in the house, wind the baby well, and put it in the pram, rock him to sleep, [hopeful face]
Thats what I used to do, do try singing/music, sometimes they stop crying to listen. If you have a safe secure back garden, you could wrap him up warm, and put him out there, with the rain cover on, sometimes, they sleep better in the fresh air. Sorry if that's not an option.
Its fine to lay him in a cot pushchair for a few minutes, while you grab a quick shower, to make you feel fresh. Have you been given advice about how to time the feeds, have you tried a dummy? holding the dummy gently to his face while he suckles it. Have you tried a sling? He may be comforted by that.
Hang a toy from his car seat handle, to give him something to look at, and give it a push every now and then. Sit him infront of the washing machine, with it on, or a tumble dryer.
This will pass dear lady, i know how hard it can be, i don't think I had more than a nap for first 12 weeks with my 3rd. Just keep trying, your doing much better than you think.