If you want a physical phone, 711 used to be the way to go. Years ago I bought a phone for $100, came with a $50 airtime credit. You have to top it up every 6 months, so it's $40 a year. But they didn't take any personal info down when I bought it and I walked out with a fully activated phone. If you pay cash, it can't be traced back to you. Plus it's cheap enough that you could trash it and walk away from it if you absolutely had to.
Keep it in your car, well hidden (in the trunk tucked away works for me). Apps etc. are good but I like having a phone.
I almost forgot, I bought a car charger ($20) so when I use it I just plug it in run it off the charger. Otherwise, it's never charged so you don't worry about it ringing at inappropriate times.