Just remember that you only have to do the "story" (HM - there is no "normal mode") versions of those primals. In other words, you only have to beat each primal once during the story. You don't have to any Extreme primals, and you don't have to do the HM versions of the first three (Ifrit, Garuda, Titan), and you don't have to fight Odin.

In addition to the trials and dungeons mentioned above, I believe that one of the early post-game quests requires you to run a lower level dungeon that wasn't required for the main story (Qarn (normal)?)

The quests are actually quite quick to burn through though, and the vast majority are either pure dialogue or fetch/courier "go from point A to point B" quests. Getting to the end before Heavensward isn't really that hard. It should only take you a few hours, honestly, and most of that is waiting in queue for the trials/dungeons (or running the trials/dungeons).