Yes, it most definately could be PND, you sound like me.
Firstly, you would be amazed at a mothers capacity to survive with very little sleep. But its so hard, i know. My PND was missed and as a consequence it took, is taking much longer to get sorted - there are other issues too. But was all brought to a head by PND.
I too had a traumatic birth and it played on my mind, i also had health concerns about my DD and i wouldnt let anyone see her for two weeks in case she died
I would talk to your HV as the first port of call, then your GP. They will probaby do a questionnaire to assess if you have PND. Mine definately manifested as anxiety so i know how you feel.
The upside to this is, there is help out there, lots of it - you might benefit from medication, especially as you have caught it early - it certainly helped me. Not sleeping tablets but medication for anxiety which of course interferes with sleeping.
Do please ask for professional help - dont suffer in silence.
It is frustrating for partners too - but honestly, i used to want to Kill my partner for saying "pull yourself together" -(i actually did try once ).
It is of course perfectly normal to feel overwhelmed and exhausted when you have a new baby. It might actually be a good thing though taht the family are stepping back now - it will help with your bonding with your baby.
Get help - you should be enjoying this special time.