Energy

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.

Speak In-Character

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 Allies

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.

The Adventurer's League FAQ states that characters with sufficient experience must advance at the next opportunity.

When do characters advance in level? [...]

Characters with sufficient experience must advance at the next opportunity: the end of an episode of a multi-part adventure such as D&D Encounters or a published adventure, after a D&D Expedition, or after a long rest. The language indicating that a character can advance after a session is incorrect and will be corrected in future editions of the Player's Guide.

(Emphasis and excerpting added to highlight relevant bits.)