I'm listening to Eye of the World(first time reader), and there's a part where Moiraine was explaining to Nynaeve about touching the One Power for the first time, and the reactions to it. She says that the first time you touch the Power is probably something you needed, like a branch reaching out or healing someone, but about 7-10 days later you experience the reaction of touching the Power(fevers, headaches).
This lead me to think about the groups run to Taren Ferry. It's said Rand felt a tingling, and he was urging Bella to run, to keep up with the other faster horses. Then when the group got to Baerlon, he was afflicted with headaches. So, the question is, how long was the run from Emond's Field to Baerlon? Was this meant to be an indication that Rand could touch the One Power himself?