I can relate to much of what you said about exhaustion, Loneliness and drinking:these were some of my reasons for drinking too. I agree with ruby and others who have noted that the exhaustion of early sobriety passes and eventually is replaced with more verve and energy. How long that takes is individual, but in reading here it does seem to kick in for folks.
One thing that I have realized is I cannot run myself ragged--at least not if I want to stay sober. You have your hands full,it sounds like. Is there the possibility of getting some support and assistance with some of the childcare and housework? Could be especially useful in the early days of sobriety. Does your girlfriend support your efforts to become sober? If so, maybe she could temporarily take over some duties or you could hire help if that's a possibility? I know it can be hard to ask for this: my partner works ridiculously long hours and I hated to ask him to do more. But it helped me tremendously and has made me a better partner in other ways.
Sending support and peaceful thoughts