I told you I do it by memory in answer to what you asked.
And how do you handle distances once characters start moving in directions that aren't straight lines away or toward each other?
Everything is straight lines, or nothing is. Whatever, it's no harder. I don't need maps, pencils and token, because I can already figure it in my head. Having paper with squares or whatev' won't make it better nor worse, unless you need because you prefer it that way (or the reverse).
Player A and B start 30 meters south of NPC 1. Player 1 moves 10 meters north. Player 2 moves 10 meters east. How many meters is player B from NPC 1 and Player A?