There's a number of different things going on here.
As a family, you are stretched too thinly.
There are a number of problems with the way that responsibilities are allocated within your household.
You need to make time to collect your medication.
I'd separate them out and address them individually. Call in sick, collect your meds and sleep as much as you can.
Discuss with your partner what a fair allocation of responsibilities would look like in your house. If you're both working, he should be doing half the drop-offs and pick-ups, and his share of childcare, organisation and chores the rest of the time. The way you deal with him not waking up in time is to leave for work yourself and leave him to it. Natural consequences and all that. He'll be late and it won't happen again.
If, having put in place a fair allocation of responsibilities, you are both still overwhelmed, it would be a good idea to look at one or both of you dropping some hours to give you some slack in your lives.
Finally, the behaviour you've described doesn't necessarily sound abusive as a one-off, but if it is part of a pattern of behaviour, then yes, I would say it amounts to domestic abuse and you need to think about exiting the relationship. If your partner's response whenever he's short of sleep or in a stressful situation is to shout or scream at you, that's not ok. Even if a one-off, he owes you an apology - it's not acceptable behaviour.