I recognize The Puzzle That is Me as belonging to felisblanco and I'm going to assume if you read that piece from her, you must have also read The Doors of Time (http://felisblanco.livejournal.com/1090487.html), her masterpiece in my opinion. But for some reason, you hadn't, I think it might fit into the 'socially awkward' category.

This one has an antisocial Jensen in it and j2 meet in extremely unusual circumstances. I'm not sure you like these types of stories but I found it entertaining and different.
Reanimated (https://archiveofourown.org/works/151617)
summary: In which Jared manages real estate in Heaven and Jensen works as Hell’s attorney in Death Trials. Extreme ambiguity and a distinct lack of beer push Jared to accept a different kind of job in a hotter, more southern locale. The boys meet. Hi-jinks and grocery-stealing in Hell ensue. Featuring saints on book tours, love on trial, after-life bureaucracy, and sin. Surprisingly not as cracktastic as it sounds.

This is also in a weird setting. Talking Through a Different Face (https://archiveofourown.org/works/229144)
summary: When single dad Jared Padalecki is recruited under desperate circumstances to become the “normal” face of a house full of misfits (including a vampire, a voodoo priestess, a man who has modified his body to look like a jungle cat, and an all-too-naked Misha Collins) he unwittingly enters a world where magic exists. He thinks his greatest problem is figuring out his surprising attraction to Jensen “Totally Not a Warlock” Ackles, but when his suddenly super-powered six-year-old daughter goes missing, he needs to learn to trust his newfound family to get her back before anyone else is harmed.

antisocial!Jensen Monster (http://redrum669.livejournal.com/7862.html) by redrum669
summary: Jensen lives a reclusive life and acts like a jerk for a reason. Jared sees his chance in uncovering it when tutor sessions born of desperation give him the opportunity to talk to Jensen in private. When he falls for Jensen, though, secrets of the past don't seem all that important anymore. That is until life-altering news have Jared wondering if there's more to the proverbial skeleton in Jensen's closet than he ever thought.

awkward!Jared Disaster. (This is a Good One.) (http://aisalynn.livejournal.com/120738.html) aisalynn
summary: Jared, a college drop-out, unlucky in love and a self-professed "loser" finds himself at the end of his rope. His boyfriend, Misha, has ditched him at the last moment, right before the family Christmas dinner Jared'd planned to bring him to, and thanks to an untimely phone call from his mother, Jared's family thinks Misha is still coming.

A series of uncomfortable moments later--including awkward phone calls to his exes, a terrible trip to the mall with is best friend where they were escorted out by security, and a run in with a schizophrenic who thought Jared was trying to take him into a secret government lab--and Jared has Jensen--an angry, former lawyer who's dedicated his life to charity work--handcuffed in his car, blackmailed and forced into pretending to be Jared's boyfriend. As the holiday stretches on and the charade he's involved in becomes less and less a lie, Jared is only sure of one thing: he should stop listening to Chad.