Going by the original title, as I am far less familiar with the remake, normal encounters do scale with you but boss encounters do not. As I recall only two fatestone quests require fighting dragons, but the other quest can't be started until very late in the game anyway so you don't have to worry about that one. While the fatestones are nice they are not required for completing the game, so don't get too hung on obtaining them super early, especially the one you found as it is one of the ones that should always be available.