Do I have to break a full property set to pay if I owe money in monopoly deal?
TL;DR - Yes you will have to break a full set.

Looking at the rules here regarding payment of rent it says


  If you have no cards in front of you to pay with, you don't pay at
  all.


So, if you are able to pay the rent you must do.  As you have a full set in front of you, you have cards.  Therefore you must pay.

There is nothing in the rules that says once you have a set it is 'safe' somehow and can not be broken.