I could have written this post myself! (Except mine was waking up even more if you can believe it, and still does some nights)
First of all, don't worry about anything during the week except taking care of the baby and yourself. Try to work in a shower every day (even if you and your husband have to get up half an hour earlier; for me it's worth it for the clean feeling) and do something little for yourself like brush your hair. Feeling human goes such a long way.
The extreme sleep deprivation you have will, over time, lead to all sorts of physical, mental, and emotional problems, including depression. There is nothing wrong with you that a good week of solid sleep wouldn't fix, if you could get it. I went through the same thing, and really still am. It's a struggle every day.
Make your husband take the baby at least one night a week, on a weekend. Pump a bunch beforehand if you're breastfeeding (and I know it's hard to find the time even for that), give your husband a bunch of bottles and then lock yourself in the bedroom with some loud white noise. Your husband is only to come get you in an emergency or if something's really wrong with the baby. Or tell him to wake you up for feedings only, so you'll at least get a few hours of sleep. Do not feel guilty about making him do this! If you don't get at least a tiny bit of real sleep, you'll just get sick and be unable to take care of the baby anyway.
And if you're like me and need the house to be relatively clean in order to feel good, try making your husband watch the baby while you clean on the weekend (or have him clean :)
Just because your husband has a day job doesn't mean you should have a day job AND a night job AND a weekend job! You need your rest too. It sucks that you don't have friends/family nearby to help out, but that just means you and your husband need to have a supertight partnership.