I think you possibly need to lower your expectations a bit, sorry.
Barely being able to get showered/dressed is often the case with a lot of babies this age, never mind being able to work!
Feeding all the time can be due to a growth spurt/wanting to be close to you which is completely normal or it can be due to a milk transfer issue such as tounge tie so it may be worth getting that ruled out.
Not being able to put down is normal, have a read about the 4th trimester.
A sling is good-you can feed in the sling and can at least make lunch, etc.
Can your DH make your lunch the night before/before he goes to work then you can just microwave.
Make sure you've got loads of drinks and ready made snacks in so you are always able to get something to eat (you get used to carrying baby on boob to kitchen and back!) or get them ready on the couch before.
Depending on what time DH leaves for work, get showered/dressed before he goes so at least one thing out the way.
Get a bath when he comes in from work when he's holding DD.
Make sure DH is doing all the cooking,cleaning,nappy changes, etc when he's there.
Often that age is just about survival, you'll soon find your own little ways that work to make things easier.
Try to enjoy sitting and feeding (get comfy on the couch with the remote) or have a rest in bed (feed lying down). I know it probably doesn't feel like it now but it goes so fast and in a few months you'll be missing having a little sleepy baby on you.
This little stage doesn't last forever.