@Chronicallytired Have you actually been diagnosed with ME/Chronic fatigue? Because I was where you are (non - child related) and after years, I was diagnosed with not one but 2 sleep disorders. I wasn't aware that I had them and would have told you that I slept well, because I had had, what I thought was an unbroken nnights sleep! Treatment for both has given me back my life.
I understand entirely, how difficult it is to stay awake, even without a baby or small child in the house, and unless you have been there, people won't understand there is a difference between what you are dealing with and 'normal tiredness.
Of course, as everyone else has pointed out, having a baby is always going to make you extra tired, and there isn't much you can do about that bit, but even if you have been diagnosed with Chronic fatigue, I would be asking your GP for a referral to your nearest NHS Sleep clinic. You will need to be careful and tell them it has always been a big problem- well before the baby came along - otherwise they will just assume that you are extra tired, due to the baby
I do think that at the moment your partner sounds as if he's doing his best, though when He is working full time. And your baby will stop screaming eventually, but maybe see if you can work out what is making him screaming.
Good luck