Not to take away from the story, because it's downright awful, but in Ukraine, most phones use pay as you go burner sims. That's what I used when I went there. I picked a sim card up for $1.
My point is that fortunately, most of those phones are not tied to you and your social security number like phones in America.
Changing SIMs might be useless, since the carrier can track phones by their unique IMEI ( https://en.wikipedia.org/wiki/Imei ) and when you use another SIM on the same phone to talk/chat to your friends, 'gov sigint' can use that information to get to you.
My point is that fortunately, most of those phones are not tied to you and your social security number like phones in America.