That seemed to be the implication in canon too, but I don't see why that should be. Even though it is stated dragons cannot live without magic, they seem able to use magic even without their Eldunarí nearby. For example, Glaedr used magic at Gil'ead when Oromis was wounded, while his Eldunarí was in Feinster at that time, something like fifteen hundred kilometres away, so it seems unlikely an Eldunarí is an absolute prerequisite for using magic and thus for living.