We are a 3 person family so don't fill the dishwasher daily. But we do use it - it's full in 2-3 days, and we run it as needed. And fill it with pots if we need to, or just run it half empty sometimes. USE THE DISHWASHER!
Get a sling. Baby is close to you but you still have arms free for some things, like a load of washing etc. And get a bag as well - so you can carry things with you easily to return to right places etc. Just any kind of bag that will go over your shoulder and carry a few bits.
Put the baby down, let them wake and settle them back again in the crib. Take it slow, and initially just concentrate on letting the baby get used to sleeping in the crib rather than always being in your arms. Once asleep, the first few times catch up on yourself - a nap, a shower, read a newspaper or a chapter in your book etc. Then look to do one small job at a time - so brush the floor, or tidy the table, or start dinner preparations (find ingredients or peel veg etc) to be able to just turn on pots later.
Things like making the bed can be a game with baby involved. Settle baby in the middle of the bed, lying down between the pillows or propped up supported by them, and make the bad, including flapping sheets around and making a wind, and enjoy it. Have a few tickles and snuggles too.
And don't have silence in the house. Have the radio on in the background (or the tv), run the hoover/dishwasher/hairdryer etc as needed, chatter to the baby (or yourself). Partly to keep in touch with the world, and enjoy your time, and partly so the baby gets used to noise and is not disturbed from a nap just because you turn on an appliance to do some work while they are asleep.
Get out for a walk with the baby in a pram every day.
And while your DH may work long hours, he should also be taking up some of the slack as your job at the moment is to nurture the new baby and recover. So getting the shopping on the way home, or folding the laundry at night, or doing the washing up when he gets in and has eaten his dinner etc.