The accelerator pump just squirts a bit of gas into the intakes when you open the throttle from about 1/8 open to about 1/2 open. It can't stick open and cause overflow - you get one squirt then you have to close the throttle to fill the pump up again for another squirt.
Overflow is most likely to be from gummy, dirty, or worn out float valves in the carbs. They are fairly easy to service or replace, the trickiest part is getting the float hinge pins out without breaking a post off. The pins should slide out freely but corrosion and gum can make them very stuck and the posts get corroded and fragile if the bike (and carbs) has been stored poorly.
As has been mentioned many times the actual overflow tubes - brass pipes sticking up inside the carb bowl - will split from freezing if the machine is stored outside in freezing weather and there happens to be water in the carb bowls. Then they leak. But a sudden leaking after doing work is probably float valves. Dirt in the gas tends to clog the valve and hold it open, possibly you stirred up some crud in the tank? A fuel filter in the carb feed tube(s) is good insuarance against this problem.
