Drummed out means that you will be forcibly removed from the military camp. When soldiers are drummed out, it is often done as a disgraceful march out of camp, to the steady beating of a drum. Remember, you are subject to the same punishment as the soldiers here in camp, but are much less tolerated.
Being drummed out is the least of your fears, should you commit more intense crimes than insolence. Get caught stealing, and you will definitely be thrown out of camp. You know many women who act as prostitutes in camp, but Washington carries a reproachful eyes towards lewd women that might corrupt or distract his men. You know of one woman giving her services to men, who was found out, given 100 lashes, and additionally drummed out of camp.
Find yourself in something deeper, something that could be seen as treason or heavy theft, and you will not face the drum; you will face the gallows.