I noticed that if you have a save file, fail the conditions to an achievement, reload the save, and then complete it properly, you do not gain the achievement afterwards. You must restart the entire mission over again to try for it again. This includes completing the hundred year war (and thus completing all of the campaigns for me).
I started the last mission for the French, and got one of the cannons killed, so I loaded a previous save, and started it again. I completed the mission and did not allow any of the cannons to die, but did not achieve The Hundred Years War, Careful Cannons or Age of Empires IV Campaign achievements.