We're aware of two certain sources of procuring the Dragon Shouts - ancient walls and the dragons, themselves.
there would be these scattered walls across Skyrim from which the dragonborn would learn the shouts
These shouts, though, aid us only in combat. So, how do we further the implementation of the language while ensuring it isn't obtrusive?
Firstly, it could be used in ancient manuscripts and scriptures for augmenting our lore. The GI article hints at this but the usage is only limited to walls.
Every ancient wall you encounter carries an ancient legend.
With puzzles being confirmed it could be used for tablets which we may obtain or recover while we scour the landscapes of Skyrim. These tablets could assist us in deciphering puzzles encountered in dungeons. Or, it may be used in manuscripts which entail information about some hidden treasure waiting to be discovered. These could also be obtained as rare rewards from a successful dungeon crawl. Who knows? There might be someone out there looking for these manuscripts to further his cause.
Additionally, since dragon shouts are so powerful, we could possibly chant them to enhance the effects of potions or poisons; provided you have the requisite knowledge (read: words from Dragon Shouts.)
Perhaps, it could be implemented to perform rituals. I remember fairly well that a Mages Guild quest from Oblivion required you to observe a summoning ritual. It would be interesting to perform one yourself by deciphering a codex or a short manuscript to summon the Daedra lords.
Lastly, we could use it to have conversations with greater dragons, albeit that seems highly unlikely.
Thoughts?