I've been doing a no sex playthrough and the riots just started, I was almost done with the story too. I'm assuming they'll happen even if I stop going for walks so I wanted to ask if I could stop them or if I just need to sleep through them.

It is a Fixed event, you can not avoid it. It usually happen on Day 30.

If you don't care about Annalie or Shanice and PriestBot, you can just sleep through the riot for 5 days and the riot will stop.