"I think he'd go mental if I just dropped DS on him."
It's your DS I feel the most sorry for here, when parents get shitty at each other because he's been 'dumped' on them. How must that feel?
"He has always (even before DS) been very clear Saturday is his time. Any time I bring it up he just says you know that's my time. Stop making a fuss."
This is someone clearly telling you three things:
One: my needs are more important
Two: you're being the problem here
Three: it's never going to change
I haven't told DH yet but did remind him to take DS with him. He said yeah we'll see. So have to see how that goes.
He said "yeah we'll see" sounds more like he's planting the seeds of doubt to prepare you for it to be another resounding "no".
OP, hate to tell you but he's not going to change for you here. No wonder you're burnt out. You're a single parent already.
I have a medical condition which demands rest during the day or I am very ill. I would push myself to the limit and then ask for minimal rest, and my exH would do the sort of thing yours does. Sometimes he would even buffer off to get a quick Kip himself, leaving me to the DC alone, which was even worse medically than not getting my rest!
I waited years for it to change and it never did. I don't think yours will either, and the collateral damage is children feeling unwanted and therefore unloved.
For a few years I sought help externally, like my family and even paid a nanny to take DC for a while so I could rest. That was particularly difficult to swallow when exH was in the house too. Ironically he would actively co-parent with the nanny and stop as soon as she left.
I'm sorry you're facing this, it's pretty shit. Find ways of looking after yourself and your DS, and make sure neither of the two of you are neglected. But I'm afraid you're not going to get help from the place it should be offered.