To encourage more role-playing, ask questions. This is easiest when you're the DM - "OK, you kill the orc! Please describe how you do it" - but you can also do it as a player - "Hey, what did my character see yours do when you rolled that critical hit?"
In my experience it's best to ask for these moments of player narration after specific, infrequent game events whose outcome is already established in the rules. To address each part of this:
Specific is because you want the whole group to get in the habit of expecting to describe certain cool events. If you can make it a "rule" - like 'the player who lands the killing blow gets to describe the kill shot' - then it's easier for everyone to learn the rule and follow it automatically.
Infrequent is because, as others have said, people come to play for many reasons; you want to use player descriptions of what they do as a spice and a teaching tool, not something that has to happen every time a player wants their character to do something (which some will experience as an onerous burden). It's best if the events you're describing are significant moments, which are also infrequent.
Tied to game events is because people who aren't familiar with role-playing games are still familiar with other kinds of games. In a scene where character conversations are taking place without obvious stakes riding on them, some players won't get into it because they don't see where the game is here. By putting the game part first - "cool I rolled the dice and succeeded or failed" - you can then introduce the role-playing part - "oh yeah, I see how it's even cooler when I can imagine and share exactly how my awesome guy was awesome, and watch that get picked up by other players and become part of the collective imagination of the scene."
Outcome already established is because people feel ripped off (and turned off of narrating their character's actions) when they come up with a cool description of what their character is going to do, and then the dice say 'no, actually you fall on your ass.' Good moments when the dice have already spoken but leave room for narration are death scenes and becoming bloodied (your character or your foe), critical hits, and critical failures. I think this last one is especially important because failure can make a player fall out of love with their character. As DM I like to say "OK, so your character is a legendary hero, normally able to do mighty feats, but something went wrong here; what was it?"
Although what I'm talking about here might not seem like the thing you're thinking of when you want more role-playing, I think they're related if not the same. You want players to say what their characters are doing, thinking, and experiencing, not just what mechanics they're using; getting them to describe their actions is an important first step in that direction.
I hope they pass, and try to plot for them passing but I feel like I need a swoop-and-save always in the wings.
Ah, here we clearly have part of the problem. They're in a situation where they would fail most of the time if you weren't engineering things for them to succeed, so they should fail some or most of the time. If you feel bad for railroading them or for removing the challenge from the game by ensuring that they don't fail, stop doing that.
PCs don't need to succeed all the time, especially when the danger isn't lethal. Failure can be instructive and even fun! (Even, dare I say it, . . . "character building"?) Honor is so important in that setting that a moment of lost honor, a recognition that they aren't perfect in all things, can do wonders in terms of future character motivation. Plan for the characters to fail.
That is not to say that they should never succeed or not learn anything, though, but there are ways of doing that without having direct social-roll confrontations or even after failing at all direct confrontations.
First, don't worry about giving them a loyal subordinate with the social skills to take on the courtiers some of the time, and don't worry about fudging the rolls for him up or down if you have to . . . maybe he's talented but new and prone to rookie mistakes or getting rather old and not as fast or perceptive as he used to be. Also, don't worry about having him advise the PCs. However, his advice shouldn't just be "do this"; have him offer multiple options with pros and cons and let things play out from there. Heck, let him be wrong sometimes just so the PCs don't rely on him too heavily. (A sufficiently deep intrigue will have elements he couldn't expect, or that were expecting him. . . .)
Second, give them plots where their bumbling or social weaknesses are strengths. Have enemies overestimate them and plan for the wrong reaction from the PCs, have their bumbling somehow convince everyone that they know more than they do (spooking the enemy into making mistakes), have them accidentally round up all the right people for all the wrong reasons and someone confesses to the plot and then asks how they knew, or just plain have things come to a point where the villains are expecting some nuanced social reaction they can parry and the heroes decide to just charge in and take care of things the old-fashioned way.
There's one more option: The overestimated idiot at the center of a backstabbing circle. In social politics, this is where everyone tries to curry favor with the lord by telling on everyone else. As a result, the lord is spectacularly well-informed about everyone else's secrets and develops a reputation for being an omnicient badass when all they had to do was sit there and look stern and knowing. If your players can handle that kind of information effectively, they'll find ways of taking care of business without ever having to make a social roll they don't like.
Best Answer
Seems to me you have two areas where you want to optimise; I've run large games before (8 players in a Rolemaster campaign, I must have been mad!) and keeping all the players involved is tough, here's what I learnt from it.
Combat
Social
I notice in your question you have When we do story based adventure there are 2-3 players that do most of the talking and the others twiddle their thumbs. To me, social situations are where a GM can be the most lazy, especially with a big group.