Put focus into roleplaying, even if you're watching someone else do it. Focus yourself on what they are saying and doing, even if it's kinda boring, and project your body language and voice while you're acting in-character. Be much less high-intensity when simply describing your bonuses while rolling, or asking someone to pass the chips. This will create natural focus on the roleplaying aspect of the game.
Wherever possible, assume what others are doing is a prompt for in-character roleplaying. DM asks what you do? Turn to someone, and say 'Philius, methinks we should cross that bridge and by Everam and St George, charge those there gnolls with swords in hand. Once we have them subdued, we'll take some answers from them, so we must leave at least some alive! What say you, well-met friend?"
Even if people don't respond in-character, and instead shift it back ooc, roleplaying just happened. Keep doing it and others will soon follow.
Ham It Up
Your character is 'quiet' and 'reasonable' and NOPE. Your character is the hammiest of ham. He's a loud cliche. He is instantly identifiable as the tropes that make him up - and he defines the setting by his very presence. It's unfortunate - but humans love ham. They love it.. a lot. Be something simple and understandable and loud, and they will get with the program really fast.
You can do this by being a masterful actor and roleplayer with any character, even a non-hammy one, but it is easiest with ham, so ham I will advise. Your paladin isn't just a paladin who likes cheese and moonlit walks on the beach - he's SIR GALAHAD THE MIGHTY, SUBDUER OF THE PEASANTS, DEFENDER OF THE WOMENFOLK, AND HIS MOUSTACHE BRISTLES AT THE SLIGHTEST SUGGESTION OF DRAGONS.
'Big' traits tend to focus things on the roleplaying a lot faster. Simpler is easier for the audience to understand.
Find people who will respond to your dramatic offers. When you address people, address them first, so they respond in-character, and then immediately pull other people in. People ignore offers initially, but if something is already rolling, they'll get rolled in with it. Some people will instinctively resist roleplaying offers, for all kinds of reasons - learn to identify them too, and offer to them last, once the roleplaying scene has the most momentum.
Be good at plot
Being able to identify where the adventure is going will let you advance the plot during a roleplaying scene - which both speeds up the adventure and means the time spent on roleplaying won't cause a weak GM to not let you hit the end of it.
Roleplay during combat
'LOOK OUT, FARAMIR! THE GNOLL IS AT YOUR BACK!' 'Galahad charges at the gnoll attacking' moving mini 'faramir, and' rolls dice 'swings at it with his mighty sword.' By including both speech and roleplaying-description in amongst your mechanical actions, you partially negate the disconnect that happens during the mechanics-rich combat portions of sessions. Have to know what you are doing on your turn before your turn rolls around, or anti-roleplayers will complain your roleplaying is slowing things up if you are not clearly doing it faster than anyone else.
Additionally, being good at combat, and giving tactical advice in-character that leads to defeating enemies quickly, will give more time overall for non-combat-constrained roleplaying itself.
Occasionally, roleplay during others' turns - have Galahad shout an encouraging phrase at an opportune moment. This has to be rare, and well-timed, though - an advanced technique.
Be Heroic, or Dastardly
Again, ham. By being heroic, and roleplaying it hard, you make other people who are not roleplaying feel heroic. By being dastardly, and roleplaying it hard, you make other people feel heroic also who are not roleplaying. You're giving them some of your roleplaying energy in a way that feels good for them. Morally grey is, again, a tougher sell. Note this isn't 'good' or 'evil', it's more saturday morning cartoon than that. Snidely Whiplashi, or Dudley DoRight.
Incorporate the GM
Don't just roleplay at fellow players. Roleplay at NPCs. Treat them with importance, and give the GM offers to roleplay right back at you. All of this applies to the GM, too. Getting the GM on-board with roleplaying, especially if you can advance the story while doing so, will be a tremendous boon to your cause.
You're right, by the way. Premade adventures, split up groups, schedules, public venues, this stuff just kills roleplaying and really makes it quite hard - I literally could not design a better system to do so.
But even in those kind of circumstances, I have personally sparked roleplaying in some extremely tough crowds. You won't see a huge improvement - but even the tiniest bit of roleplaying can be a huge welcome to you if you're in a roleplaying drought, and if you play with regularly the same pool of people, you'll find people gravitating to you that appreciate roleplaying, perhaps even to the extent that people will fight to have you in their groups.
Overall, though, the roleplaying will be in many ways a simpler thing than the rare high level roleplaying you can get in a home group.
But it's certainly not impossible.
Just have the courage to keep trying and don't give up.
D&D Adventurers League is an "organized play" system.
Local stores host games (often with volunteer DMs, sometimes with paid ones). Players play point-build or array-build characters in whatever adventure is being run. Character's experience total is tracked, and players can drop-in/drop-out on a session by session basis.
Ideally, it works best if the adventure has the same 3-7 players week after week, but that's not a requirement. In fact, that's often a problem to arrange.
A given character can only participate once in a given adventure. If a player wants to play it again, they can use a different character.
From the DM perspective, the DM gets his marching orders from the store. The store either gives him the download password or gives him the adventure to run. The DM then runs the adventure at the store in the allotted time for the first 7 players who sit down for it.
Wizards keeps track of total player numbers - they do this using the DCI numbers. The store records your DCI number, and reports it to Wizards.
All your XP is tracked on forms that you bring with, not by Wizards.
The store needs to have a coordinator - they keep track of who played, and make certain the DM's have allowed adventures to run.
At present, that's a pretty decent selection. As season 3 is about to begin, all the season 1 and 2 adventures are legal for season 3 play, as is Lost Mine of Phandelver (in the Beginner's Set). So, on day 1 of season 3, there are over 30 adventures the coordinator can choose from.
The coordinators schedule the events - there are certain restrictions on scheduling - and upload the DCI numbers of those who played in those events. They also serve as a safety net - if a player or DM is acting inappropriately, the coordinator can toss them from the event.
For full credit, events are supposed to have their results entered within 24 hours of completion. Note that, for MTG tournies, that includes the winners, but for D&D, it's just who played.
What the Various Elements Mean
Encounters: Play of the "big module" for the season in its reduced form, covering levels 1-4, on Wednesday nights at a FLGS sponsored public game. These modules are coded DDEN, and are released 1 per season. Tends to be about 10-20 weeks of play, depending upon DMing style and player behavior.
Expeditions: play of any of the expeditions modules at an FLGS sponsored public game or at a convention. These modules are coded DDEX, and there are more than a dozen per season. They tend to run 3 to 10 hours of play each.
Casual Play: Any other store or club sponsored public event use of the DDEX and/or DDEN modules, or of the hardcover big adventures, that is being reported to Wizards. Can include store reported home play, and play of the hardcover versions in store.
Epics: convention modules not available to the stores. They look to be set for 4-8 hour play blocks. The modules are coded DDEP, and while generally unavailable, leaks have happened.
Home Play: a limited exception to the program, certain modules can be played at home without reporting. It is only allowed for the hardcover modules (At present, Hoard of the Dragon Queen, Rise of Tiamat, Princes of the Apocalypse, and Rage of Demons) and the Lost Mine of Phandelver boxed set.
Note: Some of these definitions are taken from non-public sources, namely the instructions to stores on how to report events. The rest are from the Season 3 Player's Guide.
Benefits to Store
Wizards cuts stores discounts based upon total DCI number using events - Magic Tourneys and Open Play events, as well as D&D Encounters, Expeditions, and Casual Play events.
They also expect that Encounters brings people into the stores. Further, some casual players will show up to play, and decide to buy dice, minis, snacks, or even rulebooks.
Benefits to Players
You can almost always find a D&D Encounters or Expeditions game to play in.
You get to play with people before you decide to invite them to your home game.
The adventures are in fact pretty good.
Because there are numerous pregen characters, plus the option to use characters built by the basic rules (which are free online), it's a chance to have new players find the game by showing up and learning to play at the FLGS.
There also is the opportunity for community awareness. Having D&D played in public really helps reduce the stigma of D&D. When people see it is something that can be played in public, not just in the basement, it helps dispel a lot of the myths.