Do I need to sign a travel contract to book a hotel ticket through a travel agency?

No need. You book a plane ticket through a travel agency, but the travel agency still has to go to the ticket platform to book it. All subsequent problems (delays, flight cancellations, etc.) are borne by airlines, which have nothing to do with travel agencies and do not need to sign any contracts with them. All you have to do is get a paper ticket at the travel agency. If you want to go to a travel agency to handle travel matters, you must sign a travel contract.