This is one of the reasons in my opinion that governments are barking up the wrong tree when they want us to retire later and later.
It sounds to me as if you husband is finding his working day far too tiring. Do see if you can get him to the doctor, as there may be a problem that can be cured quite easily.
What time of the day at weekends and holidays is he awake and least tired? Try if you can have a conversation about his tiredness at the time of day he is brightest. Tell him you are worried that he is overtired, perhaps suffering from a vitamin or iron deficiency.
To me he sounds so tired that I doubt good sex will help, as he is probably just too tired.
Ask if there are problems at work, or anything else worrying him.
If you can't get him to the doctor or to discuss the problem, because he just does not see it as a problem, then I am afraid you will have to build a life for yourself up, while he is asleep, but try all the other options first.