Sure, they're on a Ring Quest, but at the end of the day, the whole Lord of the Rings series is all about overcoming monsters. There is a literal monster, Sauron, and the moral "monster" that Sauron symbolizes: lust for power (which leads to evil more generally). The Ring is the embodiment of that desire for power; it corrupts everyone because—haven't you heard?—all power corrupts.
So that's the initial situation of The Return of the King: there are monsters out there, and our heroes have no choice but to finish them off. At the same time, Frodo is in a pretty bad place in relation to both monsters at the start of the novel: he is being held captive in a tower with two hundred orcs, which is about as powerless a position as you can find. But once Sam rescues Frodo and gives Frodo the Ring again, Frodo gets some of his power back: Sam's love gives Frodo a weapon against Sauron that Sauron can neither anticipate nor defend against. Still, there's that symbolic monster, and it's sure to rear its ugly head in Frodo's own soul sometime soon…
The "dream stage" is the part of the plot when the monster still seems far away. In the first three chapters, Sauron seems both distant (because we never see him physically and because we know he is distracted by the war against Aragorn and the men of Gondor) and all too present. After all, we see signs of Sauron's activities all over Mordor: bands of orcs marching, hordes of slaves being whipped, and the Nazgûl flying regularly overhead. So as Frodo and Sam approach Mount Doom, we are constantly aware of Sauron. But Frodo's more immediate trouble seems to be himself and his temptation by the Ring.
As Frodo's struggles against the Ring get more intense, his physical state grows weaker and weaker. The dude can barely stand up. After an orc slave driver mistakes Frodo and Sam for mini-orcs and whips the two of them into a forced march, Frodo collapses like a dead hobbit into a ditch once they escape. He gets up and keeps going (he's a trooper like that), but only with great difficulty and lots of help from Sam. Obviously, Frodo's will is strong, but his body is deeply injured. How can such a physically weakened hobbit stand up against the might of Sauron, caught up in his Ring?
The nightmare stage of the plot is the moment when the final fight begins and all of the odds seem to be in the Monster's favor. When Frodo reaches the Cracks of Doom, he can't resist the Ring's call any longer, even though he has endured long enough to bring the Ring right into the center of Mordor. So in his battle with the Monster, Frodo eventually loses (hey, we're just being honest). Certainly, at this point in The Return of the King, all of the odds seem to be in Sauron's favor.
But just as all seems lost—as Frodo holds up the Ring and claims it as his own—the mercy Frodo displayed in The Two Towers pays off. Gollum, driven crazy by envy and desire, hurls himself at Frodo. Unintentionally, Gollum saves the world (proof that fate works in mysterious ways). By biting off Frodo's finger at just the right time, Gollum becomes the instrument that frees Frodo from possession by the Ring. Once Frodo sees that the Ring Quest has been accomplished, he can rest. There is a lot of final wrapping up left in The Return of the King even after the Ring is destroyed. But at last, Frodo is free to leave Middle-earth to recover from the mental, physical, and emotional agony of defeating both Sauron and the evil that Sauron represents within Frodo himself.