At the front of the carb, just below the float bowl, will be found two idle cut-off solenoid valves, each angling inwards. Carefully pull off the electrical connectors from each valve, and clean the pins and sockets. Corrosion accumulates over time and eventually raises the contact resistance enough to cause one valve to close. Just outboard of each valve is an idle mixture screw. Exercising them, i.e., 1/4 turn in, 1/4 turn out, may also help.

Take this step first, to insure that it is not the problem. You may very well be correct that there is an internal dirt problem, but this step is easy.