In Chapter 24 of the Eye of the World, of the Wheel of Time series, Rand is acting weird while on Spray, Domon's ship. He's up in the masts and making everybody think he is going to fall down.
He's acting just like somebody who touched the source on their own as explained by Moiraine some chapters earlier. Obviously, he is not channeling at the moment, so when did he channel before? When escaping the dream?
Okay, third time's the charm :) It's when they're running from the Trollocs and wind up on Domon's boat. The boom comes loose and hits the Trolloc. Later one of the crew gets in trouble, but he swears he tied it securely.
TL;DR: Rand first channeled while fleeing Emond's Field, in order to help Bela run faster. However, it is his second use of the Power that causes the odd behavior on the boat.
According to Moraine at the end of EotW, she suspects that Rand first channeled when they were fleeing the Draghkar:
The Aes Sedai nodded. "At Watch Hill, Bela had no need of me to cleanse her of tiredness; someone had already done it. She could have outrun Mandarb, that night. I should have thought of who Bela carried. With Trollocs on our heels, a Draghkar overhead, and a Halfman the Light alone knew where, how you must have feared that Egwene would be left behind. You needed something more than you had ever needed anything before in your life, and you reached out to the one thing that could give it to you. Saidin."
This event occurred in Chapter 11 ('The Road to Taren Ferry'). As the chapter begins, Rand is worried that Bela - Egwene's mount - will not be able to keep up with the rest of the group. Apparently he unconsciously did the trick of wiping away the horse's fatigue so that Egwene would not be caught by the Draghkar or Trollocs following them:
If Bela fell back, he would fall back, too, whatever Moiraine and Lan had to say about it. Back where the Fade and the Trollocs were. Back where the Draghkar was. With all his heart and desperation he silently shouted at Bela to run like the wind, silently tried to will strength into her. Run! His skin prickled, and his bones felt as if they were freezing, ready to split open. The Light help her, run! And Bela ran.
As Moraine explains to Nynaeve in Chapter 21, a wilder's untrained use of the One Power results in several various symptoms:
"You felt nothing special at the time, but a week or ten days later you had your first reaction to touching the True Source. Perhaps fever and chills that came on suddenly and put you to bed, then disappeared after only a few hours. None of the reactions, and they vary, lasts more than a few hours. Headaches and numbness and exhilaration all mixed together, and you taking foolish chances or acting giddy. A spell of dizziness, when you tripped and stumbled whenever you tried to move, when you could not say a sentence without your tongue mangling half the words. There are others. Do you remember?"
Once the group reached Baerlon, Rand appears to experience these very symptoms - first the headaches & dizziness, later the giddiness & foolish behavior - ultimately resulting in his fearless face-off with the Children of the Light in Chapter 15:
The tingling filled Rand; the heat had grown to a fever. He wanted to laugh, it felt so good. A small voice in his head shouted that something was wrong, but all he could think of was how full of energy he felt, nearly bursting with it. Smiling, he rocked on his heels and waited for what was going to happen. Vaguely, distantly, he wondered what it would be.
This behavior in Baerlon ultimately only lasts for a few hours, passing without any further explanation yet leaving Rand - and Mat - wondering at his own reckless behavior.
Later, while boarding Bayle's boat, he again channels to move the boom and knock the pursuing Trolloc overboard. It is this second use of the Power that results in the odd behavior you describe while travelling downriver.
