Hi Crazy,
No great solutions I'm afraid but just wanted you to know you're certainly not alone! I'm sitting here with my 5 month old draped over my shoulder....I know exactly what you mean about assuming I must be doing something wrong. Especially as this is my DC2, and his brother was unfortunately exactly the same!
In my experience, moving to a later bedtime doesn't really help alas. There are lots of sites out there which explain baby sleep patterns, but essentially their longest period of sleep is the initial one (your 7-12 stretch) and after that they wake up more frequently.
Occasionally are you able to go to bed when your DC does, ie at 7pm? Anti social I know, but then at least once in a while you get a 5 hour stretch. Which in our state, makes you feel quite high!
Also do you have a DP/DH on the scene who could maybe give a bottle of expressed every so often to give you a rest? (I'm being hypocritical here, because I don't do this- I do all the night feeds for mine. Oh and my DC wouldn't take a bottle until 2 week ago, the little darling, but that's a whole other topic)
During the day, sod the housework etc, just do what you need to to get by. Lie down when your baby naps, even if like mine it's only for 30 mins at a time at the moment. Even if you don't sleep, you're resting.
It does (eventually) get better as they get older and bigger, I promise. I know that doesn't help much right now when you're in the middle of it. Sometimes I am almost crying with tiredness at crazy o'clock, and you genuinely think "I can't do this".....but you have to, and somehow you do.
Thank god they're cute, is all I can say :)