At least, for Rand. I wouldn't choose a dirt-worshipping heathen, myself (sorry, Deadwood marathon this past weekend).
Don't forget, Min was with him the entire winter between tGH & tDR. They spent more time together between books than Rand and Aviendha did in tSR (by the end of which the feelings had obviously started on both parts).

Technically, in American parlance, he has only really shacked up with Min, as the term implies conjugal cohabitation. He and Aviendha did not have that kind of relationship, and the slang term regarding Elayne would be "knocking" rather than "shacking."

Except that she, you know, didn't. Seriously, does anyone actually READ anything about Elayne, or do you all just see her character as a blank on which to overwrite your own stereo/arche-types? The very first time she appears in the book, she tends to the injury of a visibly common man, and complains to her mother that she is never allowed to meet any commoners. She never has a single friendly relationship with a noble until she meets Tylin, and that was only because Nynaeve broke the ice, and later Dyelin. Before that, her closest friends are the outspoken and contemptuous-of-nobility daughters of a farmer, an innkeeper and a miner from the most backwater, common and noble-deprived regions of her homeland. She is the one who is most friendly with the Aiel they first meet, and the one who advises Rand how to undercut the privileges and elitism of the Tairen nobility and better the lot of the commoners. She is the one who has hopes for his efforts while most of the egalitarian Two Rivers folk simply look at the Tairen peasants with contempt for their servile state. And when she goes to claim her throne, she makes her decisions after consulting many commoners along the way and presents her agenda as a fait accompli to the first noble she meets, and is suspicious of that same noble until she proves her loyalty with her life's blood. The good will & patriotism of the farmer and miller and the rest she interviews are what she takes at face value, while being suspicious of the cousin from her own class.

Yay! A trite cliche avoided! Actually, the "princess" falling for the "Dragon" is a huge reversal, and more in keeping with his themes of the real story underlying the fairy tale, wouldn't it?