By Plays... That's interesting. I don't think I've listened to songs in that order before. Good idea. I just tried it on my Mac (latest version of Mavericks and iTunes), and it seems to work as expected, for the play order.
There is a recent new feature called Up Next. It is a song list that you can see by clicking on a symbol that looks like three lines with a dot to the left of each line; it is at the right side of the box that shows the song's info at the top of the iTunes window. It's a song list that you can add to, like a temp playlist, with songs you want to hear "next" in order (by dragging songs to the Up Next symbol). If you are just playing from a list, like your library list or playlist, the Up Next list should just show songs on the song list in order.
Does it show the expected song list order, or the oddly shuffled order? If it's the oddly shuffled list, click the button to Clear the list (at the top of the list). Then, restart playing from your song list by double clicking on a song. Does that make any difference for this problem?