It should be common sense that after your long and hopefully successful claim negotiations with your supplier, any agreement arrived at must be documented in writing.
A handshake would certainly not be enough to finalize your agreement. Everything that has been agreed to by both parties has to be documented in great detail in a contract signed by the appropriate managers of both companies.
Factories may hire lawyers to review these contracts and my advice is for you to do the same unless you have studied law and are familiar all the legal aspects of contract law. It is essential and I speak from experience with these types of cases. Leave nothing to doubt about your agreement and also no room for multiple interpretations of what has been agree to.