You might want to complete the accepted answer to the quoted question in comments (so this answer) with some dice rolls to randomize rooms order.
As an example, you could start in a basic stone cellar, and you plan that from this cellar they can reach 2 different rooms, say a dead end and a large room. Players have two doors (I say doors but that might just be basic paths, work the same) to choose from, but what they don't know is that the path to these two possibilities isn't set in stone. When they chose a path, make the leader make a roll to know which possibility they'll get. You might say that they are only at the start of the maze, so haven't a lot of chance to already encounter a dead end, so on a 1~5 they get the dead-end and on a 6~20 they reach the next room.
This example was for only knowing if they can advance fast in the dungeon, as reaching a dead end will make them go back to take another door (maybe letting potential pursuer get closer to them ?).
Your players can also note the order in which they go from one room to another to draw (or you draw them) a pseudo map, allowing them to kind of locate themself in the maze without having a detailed map to keep track of.
For the purpose of different good paths and exits, you might want to add multiple good ways in the possibilities, not just one good one and others only dead-end.
In a room with 4 doors, you might say that the 2 left-side have a possibility to lead to exit N°1 while 2 right-side doors will instead lead to exit N°2. You might also add some hidden doors with shortcuts or your secret exit N°3.
To make this a little more than just players against dice, you might adjust odds in accordance with players' actions, discoveries, or general knowledge. You could hide some notes about the 3rd exit on a shelf in an old armory, or have a character knowledgeable about dwarf culture notice a hidden engraving on a wall/door which may help avoid dead-ends or make it easier to reach exits.
Doing this avoids the pain of drawing a detailed map, but you'll still have a lot of work to achieve, depending on how much you want to prepare each possible intersection and odds from getting from one room to another. A flowchart or other options indicated in the answer linked at the start of this answer might help you a lot with this.