How do I, a player, best encourage more roleplaying in Adventurer's League? I'm not looking to overwhelm the other aspects of the game, but am wondering about elevating roleplay (and exploration, but that's not for this question) to the same level as combat. "Three pillars," and all....

Some features of Adventurer's league, I think, make encouraging role-playing uniquely challenging as compared to the typical home game:

• DMs are tightly constrained on the use of XP: maximum XP rewards set in writing that can easily be reached with modestly-combative parties squeezes out room for roleplay-rewarding XP.
• (not so much Encounters, but definitely Expeditions) DMs are time-constrained, as the party expects to complete the narrative in x hours. So time spent on one pillar subtracts from time spent on another.
• It's much easier for a DM to run the combat-encounters in printed material than the non-combat encounters. Stat blocks are easier to internalize than personality notes, there are rules that cover most things, players are usually going to take pretty-predictable actions, &c. And in the Expeditions I've run there's more text devoted to some monsters' combat tactics than there is to the backgrounds and personalities of even the most significant NPC in a module.
• Many Adventurers League players (in my experience) are coming from recent experience with D&D Organized Play pre-5e or with Pathfinder Society. In either case the player is coming from a recent, organized experience that is weighted more toward the mechanical aspects of RPGs than I'm looking for at my 5e table.
• Another large contingent of Adventurers League players are coming back to play after long hiatuses; I think at every AL table I've played during 5e there's been at least one player 'just getting back into it after ten/fifteen/twenty-plus years.' (This group includes the questioner who inspired this question.) These players--and I speak from personal experience--often would like to sit back a bit and observe the landscape of the current RPG world before assertively role-playing like they (we) used to in the olden days.
• A third, delightfully-large contingent of AL players are coming to D&D new. For many this is their only play experience, and they're seeing lots of play that doesn't involve lots of roleplaying. They've not got much in the way of modeling.
• Shifting party composition from session to session: I'll admit that I'm more hesitant to dive deep into roleplaying with strangers than I am in my more-stable gaming groups.
• Public play: I'll admit that I'm more hesitant to dive deep into roleplaying if the MtG- and miniatures-folk are going to look at me weird than I am in my private settings.

I think that good answers, in addition to hewing strongly to principles of Good Subjective, Bad Subjective, would address many of the points above.

On the other end of things, "AL just isn't a good setting for this" isn't a good answer. I am going to play in AL, and I desire more roleplay.

I've gone back and forth a few times on whether to pose the question from the frame of an AL player vs. leaving it open to player/DM/marshal possibilities. Suggestions that address the DM or marshal more than players would be fine; mind that they should be specific to AL, not just generally-good suggestions for DMs.

Related: How can I encourage my D&D Encounters group to do more role playing?, How do I transition my players from roll-playing to role-playing?, How to introduce and encourage role playing in non-roleplayers?, How do I deal with my players not roleplaying at all?, How do I emphasize role-playing during combat?, and many others, almost always GM-centric.