We signed a contract with a property manager for our home in FL. They sold their company to another management company. Are we legally bound to this new company?

It depends:
1) If the original property management company was a corporation or LLC and the new company bought the actual corporation or LLC, then you are still bound, since the company you signed with still exists and is still in business--it just happens to have new owners--and so the agreement with it is still binding.
2) If the original property management company was not a corporation or LLC, or was one but the new owner did not buy the corporation or LLC, only the assets, then you are still bound to the agreement if the old property manager "assigned"--basically sold--the contract to the new one; commercial services contacts may, as a general matter, by assigned from one business to another.
3) However, the original property management company was not a corporation or LLC, or was one but the new owner did not buy the corporation or LLC, only the assets, AND the contract you signed has a "no assignment" clause or required your consent to assign it, then the new company could not acquire or take it over without you agreeing to let them do this; while contracts are, as stated, generally assignable, they are not if some term or provision in the contract itself bars its assignment. You need to review the contract to see what it says about assigning or transferring it, and if donin so is permitted.
Only in case 3 can you escape the contract.