Sounds like you are having a really hard time of it. I think that it's important that you put your needs first sometimes. If your partner is away during the week then he is getting a good night's sleep at that time. Explain to him how you are feeling and that you really need help on the weekends. Wake him up when the baby crys on a weekend and let him deal with the situation.
Your water is very important. Not just for your health and skin (which is important enough!) but also because the process by which your liver breaks down the fat in your body requires water in order to work. If you don't drink enough water you will be limiting your body's ability to break down fat - which isn't really the idea!
Also, you might want to think about moisturising - not just your face but also your body. Again, this is a way of looking after yourself. It will help with your dry skin and will also help your skin to shrink back as you lose weight.
I know it's hard with a new baby. Don't take everything on yourself. Ask for help if you need it. Also, remember that in looking after yourself you are ensuring that you are happier and healthier and able to be there for your partner and child. Good luck with your journey!