There are several things that may cause the tractor to die when you engage the deck. When you move the lever to the engage position, you close a contact in a safety switch and complete the circuit to the seat switch to ground the spark module and kill the engine. The seat switch is open when you are sitting in the seat, so the circuit isn't closed and the engine keeps running. Normally if the seat switch fails the engine will die instantly when you engage the deck. You should check the wire connector to be sure it is plugged on to the seat switch securely. If the connector slips off, the engine will die when the deck is engaged.
The other most likely possibility is fuel supply. Be sure the gasoline in the tractor is no older than 6 weeks. Your engine does not have a fuel pump, it relies on gravity feed to supply the fuel to the carburetor. You do have a fuel filter that may be partially stopped up. If you remove the fuel line from the carburetor, you should get a continuous full flow of fuel from the line. If you don't the filter is stopped up. If you are getting a partial flow of fuel, it can be enough to run the engine until you put a load on it.
If you disengage the deck when the engine starts acting like it is out of gas will it recover and keep running? If it will, the problem is more likely in the fuel system. The carburetor may also need cleaning, you might try spraying some carburetor cleaner through the air intake while it is running.