I have paid the old property management company, but the new property management company called and said that I would sue if I didn't pay the fee.

Sue is everyone's normal right, so the new property company can sue you. However, when responding to the lawsuit, you can issue a bill that has already paid the property fee. Because of the alternation of old and new properties, the two companies must hand over the corresponding work, including property expenses. So, you don't have to worry at all.