The sleep deprivation of the early weeks is horrible, no question. I know exactly what you mean about the anxiety and being unable to sleep properly. Also, newborns are really noisy sleepers and with dd2 in the end I had to start wearing earplugs to get any sleep. It did help actually.
I was also crap at sleeping in the day, even if dh took dd2 away i found it impossible to let go and just sleep. But at least resting is better than nothing. Maybe try watching TV to take your mind off things. Got any good box-sets?
It will pass and it will get better. I think we are programmed to sleep really lightly once we have babies!
It's easy to say but do try and let go of the anxiety. It's a vicious circle, once you start worrying about how little sleep you are getting, it makes it even harder to sleep.
How are you feeing your dd? If bf-ing, feeding in bed lying down is good, and much easier for you both to get back to sleep. If bottle-feeding, get your dh to help.
I will say it again- the first few weeks are dreadful. This time with dd2 I felt terrible and depressed and spoke to the HV-her solution was for dh to take dd2 away for a day and night and bottle-feed her while I rested. Well we did that but it was such a mistake, I couldn't relax at all, in fact got totally stressed out so it was a bit counter-productive. So am not sure I would recommend that!
If you can't sleep, rest as much as you can, and eat well. I have no idea about drugs, perhaps someone will come along who can give you better advice.
Good luck x