Not all of the following will necessarily apply, as you state that you are running a "home-brew based on 5e", which makes me think you have a substantial amount of home-brew. I have written the following thoughts in terms of 5e RAW, as much as I remember it. Please let me know if any of the assumptions conflict with your home-brew.

I'm someone who likes opportunity attacks, just putting that out there. I don't think they slow down the game considerably...unless you have a whole lot of them. So, why do you have so many attacks of opportunity?

The first thing to note is that there is no such thing as "triggering" an opportunity attack, rather there is "provoking an attack". So even if a provoking action occurs, you don't have to take it all the time. That said, there shouldn't be that many to begin with; an AoO costs one's reaction, and you only get one.

Possibly, you have too many creatures in an encounter? Consider deploying foes in "waves", which also reduces the amount of time per round with less turns, and will speed up your game that way. Less creatures = less reactions to go around = less AoOs.

But what is provoking all of these attacks? If you've got to move, one should generally disengage. Otherwise, there shouldn't be so much movement in combat (in 5e RAW). I'm trying to remember if there is anything other than movement that provokes an attack of opportunity, and I'm coming up blank. (I have a couple of home-brew triggers for attacks of opportunity, but as I said, I like attacks of opportunity). Can you share any of the details of your home brew that leads to either a) needing to move so much while in melee or b) non-movement AoO triggers?

Of course, if you're using a Marilith, you should stop that