you may be having a problem with the choke on the carburetor. Have you tried cleaning the carb with cleaner. all the linkages and springs and everything you can see. THis may not help but it is a good start. It may have something to do internally in the carb also. If you ahvent posted in teh Corolla section you amy want to so you get more views and responses.
Welcome to TN.
Welcome to TN.