The sediment bowl is just that. A place for dirt, water, etc to settle out and not go into the carb. It won't run dry, just collect stuff that is heavier than fuel.
First make sure your fuel valve is open. Next I would open the drain valve on the bottom of the carb, if there isn't a valve on the bottom of the carb you should have a plug likely with a hex head. With this open and the fuel on you should have a steady stream of fuel. If flow is slow start unhooking fittings starting near the carb and working your way back to the tank. Once you get good flow you have found the culprit and can either clean or replace as needed.
Do you have a inline filter? Also there should be a screen in the sediment bowl that can also be a problem.