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.

A Tool to Enable Consensus Decision Making

• Problem: your group fails to make timely decisions due to a consistent failure to reach a consensus
• Desired Remedy: A tool that helps alleviate this detriment to fun gaming.
• Proposed Tool: Options Identification Process and Voting Tool (see below)
• Requirements: Buy-in from GM and players on the particular voting tool that will be used.

A voting tool can resolve all four problems if your group and your DM agree to use a voting tool. We don't know the interpersonal dynamics in this group. (It matters). I will assume that you are all friends or at least on friendly terms.
Note about reality: Who the "alpha dog" in your group is may color your success in agreeing on a decision aid.

What you seek is an in-game usable form of Consensus Decision making

A generic process is illustrated by this flow chart and the previous link is a concise summary of the process that is subject neutral. (Not TTRPG centric, but process/tool set used in many walks of life).

Per your comment that the group is all adults, you could just stop here and look at the summary in the first link, and tailor your own tool. But we'll proceed ...

Apply the voting tool when you find yourselves in the dilemmas you described in the question.

First:

• Identify how many different actions or choices are being proposed.
  • If you don't identify what your options are, you can't make a decision.
  • You can die roll to see who states his case first, with the DM as facilitator.
    • (Or, and better, IMO)
  • Take turns as pointed to by the DM, as that disrupts play less.

Second:

• each player proposing an option states it, along with a brief "why" for that choice.

Third:

• With DM facilitating, you all vote on each option.
  • Each player has 2 votes available. You cannot apply two votes to a single option.
  • Use a d6 to indicate your vote, in front of you at the table:
    • 1 pip is no, 6 pips is yes.
  • A brief "why not" for a no vote is an option here
  • Rinse and repeat for each option.

DM keeps track of votes received. (as neutral facilitator).

If there were more than two choices to start with, drop option with lowest score, vote on remaining choices per above.

Fourth: Vote To Determine the Group Decision

Voting Criteria For Success:

Unanimous agreement
Unanimity minus one vote
Unanimity minus two votes
Person-in-charge decides

Pick from one of the above criteria. Your group has to agree on the level of consensus that is acceptable to all(See Social Contract comment further down).

For the final vote, I suggest Unanimity Minus One or Unanimity Minus Two.

If you end up with a hung jury due to which protocol was chosen (like Unanimous) you have two last resort options to get a decision.

1. "Person in Charge decides." You can roll for, or each night designate, someone as "person in charge" and accept their decision for hung juries.

2. Roll the dice (high wins) or flip a coin to decide between the last two choices.

Your problem statement indicates that you want the group to make decisions. The above is a time tested method, adapted for your described table, that will get you decisions.

Summary of Benefits: (to address your stated problems)
- Vote on choices to keep play moving by making decisions.
- Don't split the party.
- You'll have less wasted time.
- Each player participates in making decisions for the group when the group needs a decision.
- The GM doesn't pull his hair out.

Caveat to this answer:

1. If you are the only person at the table concerned about this, the above as a decision aid is probably doomed.
2. If the other players care, then you have something to discuss within your group and get buy-in.
3. Getting buy-in on collaborative processes like this is part of your Social Contract, which from your problem statement is not robust in your group -- at least in this area.

Experience:

Small group dynamics and decision making have been in my professional life for a few decades. I'll use an informal group example of a decision process following the same steps tailored to a different situation:

• RL example: seven men, one van, Friday night, which bar to go to? Thumbs up and thumbs down rather than dice. Same basic process, different objective, small social group dynamics.