I think your best bet is a phone and minutes you can pay cash for.
Example- Buy a new or used phone with cash that can be used with a prepay carrier. Pay cash for a prepay carrier card at Walmart / CVS / Dollar General / etc. Most Carriers ask for your name, address, etc when you call to activate a phone but they don't verify anything you tell them. Just don't forget the info you give incase you have to call customer service at a later date.
Try and keep it clean of web history, phone numbers, text messages, emails, etc. Especially if going to see someone new.
If shit gets reset and throw into salt water!
I've heard that Apple requires a court order / search warrant before they will assist LE with accessing data on a phone thats locked with a password. Use the long password. 0000 1234 is DUMB!