I think one issue here is that your DH has more children than you, and, really, probably ought to be doing more parenting. Which, if finances are tight, may mean you get to do a bit less.
Although you want to be at home with your dc, is that for your benefit or theirs? If for theirs, would they benefit just as much form being with dad, as with you. Because your dh is dsc's dad from birth, would dsc benefit more from being wiht dad rather than you, for some of the parenting time.
I suppose what I am dancing around here, is to say that as the second wife of a man with children already, you went into this with more obligations and fewer options than when both parents are first-time parents together. It's hardly a unique situation, but at the moment you are straining towards a cosy sahm scenario with just your dcs, which was never an option.
Unless your earning powers as a couple are so large you can have a lot of time with your dcs, and your dh can have a lot of time with his ds as well. Which you've said, unsurprisingly, is not how it is.
This means that something else has to give. I think a combination of:
- seeking support with dsc's behaviour - if you think he is 'acting out' in some way which certainly sounds a possiblity
- you stepping back from dsc care, your dh stepping forward - you need to earn more to cover the shortfall, but that is good for you long-term anyway IMO
-investigating options for care for dsc in a more structured setting eg after school clubs. Something which could also benefit your own children (when they are old enough). Though this has to be with full enthusiastic support from dsc's mother and dsc has to not in any way perceive it as him being pushed out. I think the timing to coincide with you going back to work would be the way to frame this.
It must be hard for you. My Mum actually told me more than once 'when you were about 7 or 8 you went through a phase when you just weren't very nice'. Not necessarily relevant to your situation, but you're not the first parent, step or otherwise, to feel that way.