Correct. The reason you "pull on the mask to activate the flow of oxygen" is that this action starts the chemical reaction in the oxygen generator that produces the oxygen. Each generator will "burn" for about 15 minutes when activated. It can not be shut off.

The masks themselves will not drop until the cabin altitude is above 10,000'. A pressurization problem doesn't mean that the cabin altitude is already above 10,000'. It just means that there is a problem with the pressurization system. There's often time to stop climbing, or even descend, before the cabin altitude gets above 10,000'.