Can your DH take over like taht more often? That's what I do when I've had enough - it happens quite often, believe me! And when DD is up at sparrow's fart I hand over again and go back to bed for an hour or 2 till DH has to leave for work. No lie, it's the best sleep I get all night.
You're not weak or inadequate EJ. You're just under-resourced. Most of us are because we try to do it all on our own.
And when you're exhausted with no hope of reprieve, of course you feel like crap. I hang on only because i KNOW it will get better. Right now we make do - I grab what rest I can, when I can and I don't expect too much from myself or my baby. The best advice I ever got with regards to sleep - and at the time I thought it was dreadful advice and hated hearing it - was to let go and give up. It took me some time to understand this advice, but it's easier to get it 2nd time round because I've been through this and out the other end before.
Do what you need to do EJ. Make your life as easy as you can and call in all the resources you can muster