I have two models that are similar to yours. You may be able to solve your problem by replacing the needle and seat, but experience has taught me that turning off the fuel valve when you are done using the mower, is the best solution. I believe that this is a good practice with any small engine that has a gravity fed fuel system.
Also, when I am done using any of my Lawn Boys, I turn off the fuel and let the the engine run the rest of the remaining fuel in the carb, until it quits.