I'm considering switching from AT&T to Verizon. Can I keep my AT&T Note 3?

[AC Question]

I've had it with AT&T's poor coverage where I live and have decided to switch to the "dark side" and go with either T-Mobile or Verizon. Even though the AT&T phone is unlocked (GSM), I have been told by both TM and Verizon that I can't use my Note 3 because AT&T uses a different internal antenna than they do and consequently, I have to buy one of their phones. Please help!