You've run into a common problem - "Party RPGs with non-Party Characters". Same Page Tool can't fix groups who want different things, and it also can't fix game design that works against it's own game premise. You have a few options:
Class Limiting
"Hey, we're playing X kind of game and these classes/types in this
game don't fit that. Can we just not use them for this game run?"
Games that usually have classes antithetical to their goals usually also have a pretty broad set of class selection, so it's usually not too bad in terms of choice limiting. The other half to deal with is the social contract of your group.
(There's also a subset of gamers out there who deliberately pick the most contrary ideas to what you state the game is about. "Dude, why do you have a Navy Seal character in our game about civilians running from monsters?" Those players are their own problem...)
Building with Limitation in Mind
"Hey, for this kind of game I want to run, these kinds of characters
will need to fit these kinds of situations. Can you spend your
points/pick your skills/build your powers to better fit this?"
This is a relatively good option - you can get stuff like "combat rogues" and such that are better designed for situations rather than splitting off. This depends a bit on the system's ability to allow customization or choices within the class system, and also lets players know up front what they need to consider with a character class build.
Non-Party Play and Strong Pacing
If you can run a game which isn't dependent upon a party structure, all those character classes generally work fine as long as their goals and concepts line up. In these kinds of games you need to be able to cut scenes relatively quickly, not spend a lot of time on wasted scenes and the players need to have good goals to aim for.
That said, usually systems that are more mechanically light work better for this than ones attempting to balance out a lot of abilities, though games like Burning Wheel or Blade of the Iron Throne can work fine for it, mostly because the basic resolution systems allow for quick play and give good goal-building tools in the form of Flag mechanics.
When a player is hogging the limelight like this, the way to deal with the situation is to stop encouraging them. The player is getting their fun by having everyone's attention focused on them (see this question for a similar situation). (This isn't a bad thing, by the way! It just means you have to make sure that the rest of the group gets their fun, too.)
Don't Reward Indecisiveness
Right now, it sounds like the entire group is (unintentionally) rewarding the player by watching when he's roleplaying, letting him stall, and otherwise paying attention to him. So the first thing you need to do is to stop paying attention to him. That sounds harsh, but doesn't have to be.
If the player refuses to make a decision in-character, the DM should simply say, "Okay. While you're thinking about that, I'm going to switch scenes to the other characters for a bit. Let me know when you've made a decision." Then the DM swaps focus to the rest of the group and ignores the player (except to answer the occasional reasonable question, such as a request to make a knowledge roll to help make a decision). This is going to be difficult for the DM, because the player will immediately do everything they can to get the attention back - asking lots of questions and demanding immediate answers, trying to RP more stalling, etc - but it's up to the DM to repeat, "I've given you all the information your character has. Let me know when you've made a decision." Then the DM must return to ignoring the player until they take action.
Similarly, the other players can help by not letting the player sidetrack or delay discussions. I've DMed and played in a number of games where one person (in- or out-of-character) couldn't make up their minds about what to do. If the rest of the group couldn't convince them to take an action in a couple of minutes of discussion, the other players would just forge on ahead, telling the character IC, "Catch up when you're ready, or stay here and guard the horses, it's up to you." Then either the player would say, "Screw it," and go along with the group, or the player came up with a reason to stay behind and sat out the following scene(s).
In cases of slow, drawn-out RP, it's perfectly reasonable for the NPCs with whom the character is speaking, to get impatient and interrupt the PC, talk over him, make assumptions, or otherwise do any of the things actual humans do when they get impatient. So if the player is giving a drawn-out William Shatner speech about "My... disease... (cough) is getting... (cough) a lot... (cough) worse... (cough) and I need...", most NPCs are likely to interrupt at this point with, "Yeah, yeah, you need the local cleric. Two streets over, third building on your right. Have a nice day!" Then the NPC will back away quickly and refuse to speak further (because who wants to spend any time around someone who's obviously sick and coughing up gross phlegm?). Note that the NPCs don't have to - and shouldn't necessarily - answer the question that the PC was actually going to ask. They're impatient and grossed out; they're going to make an assumption and skedaddle before the PC can correct them. The player will quickly figure out that if he doesn't get to the point, NPCs won't stick around long enough for him to talk to.
Do Reward Fast Movement
Rewarding the player when he acts quickly, whether by making a decision, roleplaying smartly, or otherwise playing in a more group-friendly way, will help encourage him to do those things more. For example, have NPCs be more helpful if he gets to the point quickly, or compliment him on his swift decision-making if he doesn't dawdle.
It's going to take a while - and a lot of work on the entire group's part - to encourage the player not to hold up the game for everyone. The player may respond to initial attempts to speed things along by being louder, or going even slower than usual. Ignore him, don't give him any attention, and don't let him slow down your game, and he'll eventually figure out that being a valued party member is much more rewarding than being the guy everyone is ignoring.
Best Answer
There are several ways to approach a drunk and disorderly player.
Standard drunk person handling techniques. Not really on topic for this site; Google it. Wheedle them, redirect them, you know, like you'd do with a kid. Go with it. "Roll the die, you get to take a shot!" Probably best if you're all drinking and just farting around. Some RPGs are called "beer-and-pretzels" games for a reason. I assume from the way you asked the question though that it's a "serious" game.
Call the game for the evening. "I think we're all tired. Let's wrap up and pick up here next time." Might be best. My (now ex-) wife would sometimes game with us, and would usually get smashed when she did, and would for some reason always haul off and hit me in the process. It was best to just stop before it got to that point. Also see the first additional tip below.
Bull on through. The player refusing to act translates to the character not acting, and whatever happens next is the result that unfolds in the game world. "You stand there paralyzed with indecision and, oh, the wagon hits you!" (or whatever it was that was going on). More appropriate for the ornery non-drunk, but can work here too if you're all Game Uber Alles.
Beyond those tactical at-the-table tips, you might consider:
Set some drinking ground rules next time. My current gaming group drinks sometimes, and can get loud or a bit extra jolly as a result, but everyone understands that there's a courteous limit. Expectations are important - if you invite people over to watch a football game at your house, some might expect no drinking, various people will have different reasonable expectations - some might expect light drinking, and some might expect that sloppy drunk is expected. If the group hasn't clarified, then you can't really blame deviation from what the norm is in your mind.
Double-check yourself. Now in this case there's hints that the choice the player was being faced with might be inappropriate in other contexts. (From deleted comments and edits: "It was related to a liaison of sorts. A pass would lead to unwanted physical attention, and a fail would mean failure at an important session goal.") Was it really her drunkenness preventing her from going on, or was she maybe just not communicating well being offended by the game itself? There's a line between "drunk and disorderly" and "impaired and having people test your boundaries" you may have crossed here.
The rest of the group didn't step in, which may mean that either a) they're standard passive-aggressive RPG players afraid of confrontation or b) they thought that her level of drunkenness was fine and you were just being overzealous about the game over the group experience or c) they also thought, regardless of drink being involved, that you were being inappropriate. If you had one or two, you might think you weren't being drunk and obnoxious, but maybe you were. Anyway, talk to the group and figure out not all what happened that last time, but what you all want to happen next time!