Whenever I sprint my character grunts excessively and an NPC dialogue starts up saying things like "if it weren't for (...) I'd be in a Markarth gutter" or "mind yourself in our town". Does anyone know how to fix this glitch/bug?

That was fixed in 1.3. Try sitting down in a chair. I find it clears some bugs. Are all your patches up to date?

Well, try to clear your XBOX 360 cache.

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If that not helps, load a previous save.

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Try to wait for 10 days (in-game)

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You can try to go to Dragon Bridge and kill Azzada Lylvieve.

http://elderscrolls.wikia.com/wiki/Azzada_Lylvieve (Read the Trivia section)

maybe after that, the glitch will be gone.

Something in you quest, triggers those two phrases. Those phrases are from Azzada Lylvieve. !!!!

http://oi60.tinypic.com/2psmyo0.jpg