She didn't. As I've pointed out.

Of course, the (almost) 20 year long span between NS and tEotW is canon.

But read my first post. Re-read relevant parts of the series. I think you will find that the 20 years of searching wasn't just 20 years with nonstop traveling outside the WT/Tar Valon.

Yes, that was one of her main reasons for leaving the Tower in NS. The other reasons being the Aes Sedai hierarchy (not having reached her full potential with the OP yet) and of course the search for Rand.

After Galldrian Riatin took the Sun Throne and Moiraine reached her full potential she no longer had any reason to avoid the Tower/her sisters.

She still left the Tower for long periods of time to search for Rand, of course.

On 12/2/2019 at 6:56 AM, Ryrin said:

Moirane left the Tower in New Spring with the angreal. The Tower not only wanted it back the Greens and Reds wanted her punished. The Greens agreed to let the Reds oversee it.

How did you ever get the idea that Moiraine took the angreal in NS? There is absolutely nothing in NS who supports this. On the contrary, if she did, it would be very weird that there is no mention of this in the book.

Should there be any doubt about the matter, however, this quote from the Eye of the World (chapter 8), proves that Moiraine took the angreal from the Tower sometime after NS (she is speaking to Rand about the angreal):

"We have lost the knowledge of how to make these", she said. "So much is lost, and perhaps will never be found again. There are so few left that the Amyrlin Seat almost did not let me take this one. It is good for Emond's Field and your father that she gave me her permission."

This Amyrlin was obviously not Sierin Vayu. She would never have allowed Moiraine to run off with this angreal in NS, as she had forbidden Moiraine to leave. By all probability this Amyrlin was Siuan Sanche.

Furthermore, we have this quote, earlier in the same chapter (after Rands mentions the Myrdraal to Lan and Moiraine):

"Had I known, when I left Tar Valon, that I would find Trollocs and a Myrddraal here*, I would have taken half a dozen of my sisters with me, even if I had to drag them by their necks."
*here = in Emond's Field.

When Moiraine left the Tower in NS, she was not in a position to take half a dozen sisters with her, and absolutely not to drag anyone by their necks. Neither was she going to Emond's Field at the time, and she had no clue that she would eventually end up there. So this "when I left Tar Valon" is sometime between NS and tEotW, probably a short amount of time before the beginning of the main series.

The strongest piece of evidence in my opinion is the fact that Siuan and Moiraine had been planning together for 20 years what they should do when they finally discovered the Dragon Reborn. And to think that "20 years of planning" only consisted of 2 messages is ridiculous and self-contradictory.

I found these quotes just by checking some of the early Moiraine scenes. If one were to reread the entire series, I believe one would find plenty such evidence. Some examples have been mentioned earlier in the thread, like Lan's warder cloak and his sparring with other warders.

Some evidence is undisputable, some circumstantial, all adding up to the fact that Moiraine visited the Tower at least once, but more likely several times, between NS and tEotW, reporting to and making plans with Siuan.