
You didn't sound like a terrible mother at all, just a tired one. You'll get through it, like you'll get through all the challenges they throw at us. In the meantime I'd go with Sara's 'if in doubt get them out', that's what I did, and I did sometimes question whether it was the 'right' thing to do. But it worked so I went with it, and at some point they stopped waking every two hours.
I really struggled for a while with night feeds, and used to dread being woken what seemed like moments after I'd gone to sleep. But then a friend put a different light on it by describing her experience of night feeds as a wonderful quiet time, where it was just her and her baby, noone else, no interruptions, doing something that only she could do for her baby. It didn't help with sleep deprivation, and I still craved a full night's sleep, but it did help me see those frequent night feeds in a positive light. It is tough though