I feel for you, OP, been there with my two though a long time ago now.
First, if you have a sympathetic health visitor, call her and tell her exactly how you are feeling. Their job is to be at the front line in helping new mums in this kind of situation and they can help in lots of ways- liaising with GP to get medical help with depression, giving information on local groups for new mums etc. Mine immediately rang my DH and told him I needed him to take a couple of weeks off to help me. She also arranged to put me in touch with other local mums in similar situations so we could meet for coffee and talk. Failing that, make an appointment with your GP directly, they will be very sympathetic and can offer a lot of help.
You must tell your DP how you are feeling and he must step up and help you. You need sleep as much as he does, working full time is no excuse. He should be taking over the baby on evenings and weekends to give you a break.
It is also really important for you to get out of the house at least once every day, even if only for a walk with baby in the pram or in a sling. Also, you should take priority for access to the car as you are being so isolated. Your post seems to imply that your DP managed to get to work by other means before you had the baby, so he needs to do this again. It is essential for your mental health to get more contact with human beings during the day and ideally meet up with other new mums for mutual support.
Good luck OP, reach out and ask for help, you deserve it.
I'll be thinking of you 