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.
What I have, personally, done in this kind of situation is take a close look at their character sheets, what can they do, what are they good at, and write an 'easy mode' adventure where they get a chance to use each of their abilities.
Once they come to a point and don't instinctively realize what they're supposed to do, gently ask the relevant player if they can do anything about the situation. Beyond that, don't forget to include basics regarding your GMing style.
For example, if you use a lot of traps, start with a few light and relatively harmless traps. Enough to wake them up to the idea that not everything that can harm them can be seen, should help them get them used to searching.
You can also show off some tactics that can be useful for anyone by making the monsters use it. Aid another mid-fight, flanking, readying actions, et cetera. Make sure to let the players know that those are things anyone can do.
It's fine to be a bit railroady during such a tutorial adventure, especially with newer players, but don't forget to include the roleplaying parts as well.
Best Answer
By complete and utter coincidence, combined with a little bit of incompetence in leadership, the new players show up at an opportune time. As it happens, there were two groups assigned to do the current party's job, but nobody realized it until both groups had been sent out already.
The first group is the current party, and was assigned to this task by (the king? his spymaster? his top general? his other top general? the captain of the palace guard? a scheming duke?) The second group consists of the new players, who were assigned to this task by somebody entirely different. (If you want to be complicated, have the second group be there for a different reason. Make up another reason for them to be there. The players haven't explored the entire cave system yet, so you can still add something in an area they haven't seen.) If you think that "just the new players" is too small of an adventuring party to take on this cave system, then the new players' party was once larger, but some people got killed along the way.
As for how to dispose of the characters that left the game: adventuring is dangerous. People die, sometimes with no warning beyond "Hey, what's that funny soun---". Death doesn't respect character back-stories or plot arcs or convenience; if somebody suddenly has a gaping hole in their larynx, being in the center of a web of plot won't help them. Of course, if your players have easy access to resurrection, then that won't fly. So, if you truly need a way to remove the PCs without killing them, then the second group could have orders for the PCs who are leaving, that are more important than their current task. Or an offhand comment by one of the newcomers makes one or more of the PCs suddenly run off to rescue a loved one, who was living in a city that just got besieged by an invading army.
Note, if you kill the extraneous PCs, and you make it a "deus ex machina" moment, make it clear to your remaining players that you're not going to kill them that way while they're still playing. Most players don't find it at all fun when the GM randomly kills off PCs; don't let your players pick up the idea that their PCs may suddenly be taken from them if they annoy you.