No....it's not true that they won't leave if you don't take the feeders down. Keep them up until you are SURE they are all gone. I'm horizontal to you in Kentucky and I still have one that's feeding, so I am keeping one out there until I haven't seen him for at least 2 weeks.
Migrating is an internal clock. They don't linger just because there's food. Their body tells them when they have gained enough weight, when the air is cold, etc so they know instinctively when to leave.
They fly all the way to Mexica and into Middle and South America. The ones you see next year were here the year before too. This year I moved one of my feeders from the kitchen window to the side area and this one little guy kept looking for it where it used to be. I knew he had been here last year from the way he searched.