I actually bought a few checkers books
I have a number of checkers books, but I don't think I ever got much insight into what to put into an evaluation function from them. What worked for me was watching games between kingsrow and other programs, and observing how kingsrow would lose or win.
About 'the move', I came to the conclusion that there are so many exceptions to a generalized rule that it is no help. There are some very specific cases where it is important, so for example if you have detected a certain formation, and you know that e.g. if the side with the trapped king has the move then it is a draw, else it is a loss for the side with the trapped king, you can use it for that specific case. But for a generalized eval term, not qualified with some specific formation, I do not think it is helpful.
It has been a while since I worked on checkers, but from memory some of the important terms were back rank, runaways, tempo, mobility, and many things associated with kings. King mobility, trapped kings, king control of the center, kings with limited mobility at the board edges. I think I have some specific tests for rather common formations of kings that are trapped on the king row (where they first become a king), and then a more generalized function to detect king mobility anywhere on the board. There are also a number of 'cramp' formations, where one or more pieces are more or less trapped by a smaller number of opponent pieces, usually along the sides of the board.
Many authors talk about controlling the center, but I found that early in the game, when no kings are present, this does not mean that you want to focus all your men towards the center. Center control is much more important for kings than for men.
One of the hardest things to evaluate is when it is good to sacrifice a man for an early king. If you can see that the king will have good mobility and be able to freely move towards the center, and if your opponent does not have any good chances for a king in the near horizon, then you want your eval to make the sacrifice look attractive. But if you sacrifice for a king and then your opponent gets a king soon afterwards, you probably lost.
-- Ed