L.C.
Breasts do not 'drain'. They're glands (manufacturing plants) not bladders (storage units). It is normal for babies to only nurse on one side at each feeding. It is also normal for babies to nurse between 26 and 50 times in a 24 hour period --averaging 6 minutes per feed.
Solids (of all kinds) have a lower nutrient profile than breastmilk, so every spoonful of 'other' food displaces a more calorie- and nutrient-rich spoonful of breastmilk...which may be the whole of why she's needing to eat so often in the night. So, kind of exactly the opposite.
It is abnormal for children under 4 to sleep through the night --and the medical term 'through the night' is 5 hours. In sleep studies, it has been found that even if parents think their kids are sleeping through the night, they're not. Sometimes, they stop waking their parents and sometimes the parents stop waking even though their kids are crying loudly.
Sometimes, parents get wrapped up in the ego-value of having 'the right kind' of baby (who sleeps soundly for 12 hours at a stretch) and actually lie about their kids' sleeping habits. They'll say 'she's slept through the night since she was X weeks old' but under accepting and careful questioning will admit that there has not been a single night in months when she didn't nap too late, or have teething or have a nightmare or get startled by something or was cold or too hot... the list just goes on and on. And they genuinely believe that their baby has, in fact, 'slept through the night since...' in spite of this, because all those exceptions don't count, or weren't 'really' waking. It's hilarious to watch.
Do you sleep when she sleeps during the day? I've met a tremendous number of sleep-deprived moms, but remarkably few sleep-deprived babies... you should still be sleeping when she sleeps, and even if you can't sleep, you should still be putting your feet up, closing your eyes and resting for the whole of her naps. Because she's going to be waking through the night for some time to come.