tl;dr: He was suffering from madness by the end, but his particular madness was relatively mild and easily hidden.
There are really two questions here that we can answer:
- Did Logain manage to stave off the Taint madness completely, and how?
- If not, how did Logain keep his madness in check for so long?
The answer to the first question is that he likely did succumb to madness at some point, we just couldn't tell from the outside. The symptoms are different for each person. Some Asha'man's madness is easily hidden, or subtle, or even relatively harmless. In other words, short of a POV from Logain to indicate what he's thinking, there may have been no indication that he was suffering from it.
But we get one in A Memory of Light, and one of the things he thinks to himself:
He suspected this was his madness; the terror that releasing the One Power would cause him to lose it forever.
So, he probably had begun to go mad at some point, but he clearly had it under control. This leads us to the next question: given that he channeled freely as a False Dragon, how did he keep his madness to a minimum?
The answer is that he probably channeled a lot less than most people think. Keep in mind that time in the series is much shorter that it might appear. When we hear that Logain proclaimed himself a False Dragon and started a war over it, we tend to get the impression that he spent a very long time raising armies and waging war and using The Power. But the actual timeline is much shorter.
From the time that Logain began channeling to the point he declared himself a False Dragon was about four years. We don't know how much time he spent actively channeling during those years, which has a direct impact on how fast his madness set in. Once he declared himself and began his war in Gheldean, he was captured and severed in about a year, which would have halted the progression of his madness. After Nynaeve Healed him, he again channeled for less than a year before Rand cleansed the Taint.
Five to six years is actually not at all unreasonable for a man to remain sane enough to blend in to society. There are men in the Black Tower that have been channeling much longer than Logain, albeit probably much less frequently, who have their madness at least partly in control. In short, I don't actually think Logain'a situation is really all that unusual, he simply fell on the statistically lucky side of the bell curve.