I used Cozi for years but it's kind of a pain. My ex and I have used Google calendar for years and years and it works for us. The way we have set it up is that we each have a Google account and each add each other's calendar to ours, so I can see what he enters into his calendar and he can see what I enter into mine. We'll each block off time that we're unavailable (we don't really have a set schedule, the kids are with me by default all the time). We are each responsible for linking in things like team calendars because if you link in other calendars, he won't see them through your calendar and vice-versa. I know that he doesn't link the school calendars on his account, so I do still do him the courtesy of copying events that he should know about (concerts, etc.) from the school calendars directly to mine, which makes them visible. I like that Google works on phones (it's the default for most Android phones), my tablet, computers, etc. without having to download an app, set up yet another log in, etc.
I've heard really great things about OurFamilyWizard but we haven't had to use that. My kids are now old enough (13 and 15) that they have their own phones and it's appropriate to put them more in charge of thinking through their own schedules and transportation so I hardly have to talk to my ex at all and it's awesome! You'll get there too in a few years but in the meantime, maybe look into OFW if you need something more robust than just a shared calendar - in addition to a calendar, you can use it for messaging/email, expense tracking, etc. If you need to give a significant other, grandparent, baby-sitter, your kids or anyone else access to the calendar, you can give access by role without compromising privacy of communication with your ex.
You didn't ask for this but if you haven't read "The Truth About Children and Divorce" then I would recommend it. It profiles several types of divorce and how to best set boundaries and navigate things during and after divorce depending on how high or low conflict things are. They talk a lot about high-conflict divorce so there might be some info there that helps you deal with him in a way that shields your kids without you taking an emotional beating.