I'm not a single parent but, like you, husband worked long hours. My children are all teenagers now and still in one piece so it does get better.
Lower your expectations - no one expects you to live in a show home. My simple rule was that as long as we were fed and washed we were on to a winner!
Slow cookers are a life-saver - throw everything in it in the morning and hey presto a "proper" meal is ready in the evening. Instead of your husband spending time cooking in the evening you could perhaps get him to spend an equivalent amount of time doing some housework, eg washing the dishes, laundry.
Find shortcuts that work for you, eg I used to put a load of laundry on at night then hang it out or put it in the dryer in the morning. Cleaning wipes in the bathroom and kitchen. Deep cleaning is not one of life's necessities at this point.
Online shopping for groceries. Also, there is no shame in quick, easy food. Mumsnet's favourite batch cooking and cooking from scratch can wait. Pasta sauce from a jar hasn't killed anyone to my knowledge.
Try and get out everyday - the fresh air used to send all my children to sleep and pushing the pram acts as a form of exercise.
If your baby struggles to latch on have you ruled out things like tongue tie? Don't put undue pressure on yourself to breastfeed. It doesn't work for everyone. Also, if he/she cries when put down is reflux a possibility or is he/she just clingy? Sometimes, babies just cry more than others.
Mine all hated being in slings but were fine in a moses basket or baby swing so I simply moved it around the house to where I was, eg I put the radio on in the kitchen and washed dishes etc while he lay in his basket in the doorway between the kitchen and dining room. I sang along to the music and he was happy enough to hear my voice!
My husband used to make me a packed lunch in the morning before he went to work so I could pick at it during the day. Ask ( or tell! ) your husband if he'd do the same. Husbands can't read minds - have you told him what you need help with?
Growth spurts can mean your baby wants to feed constantly - on those days I stayed on the sofa with a book, drinks, the remote control for the TV and made the most of all the cuddles.
Sleep when the baby does - it makes getting up in the night slightly less torturous. There is a reason sleep deprivation was used as a form of torture.
See your GP, health visitor etc - that's what they are there for. There is nothing you can tell them they haven't heard before.
You are doing a great job. Keep going, this stage doesn't last forever.