Is it NP-hard to play international draughts correctly?
OK, here's the reduction. Turns out you don't need planarity after all. Also, for "find a legal move", I take the decision question as "is move X legal?".

First, let's work instead with a game where pieces move orthogonally instead of diagonally. This game is equivalent (just look at the draughts board rotated 45 degrees) except for edge properties, which we will not use. We use two gadgets: merge / split and crossover. See http://www.hearn.to/draughts.pdf. We assume there is a single White king on the board to move. (No other piece will be able to capture any significant number of pieces.) It will move through the indicated corridors, capturing black pieces along the way.

First, merge: if the king enters on any of the N paths A (via capturing a black piece, not shown), it can exit at B. Likewise, if we reverse the gadget and it enters at B, capturing the shown piece, it can exit along any path A (again, capturing an external black piece). This is a single-use gadget (because the exit black piece can be captured only once).

Second, crossover. If the king enters via A (C), it can exit at B (D). It can't stop in the middle and change routes, because that would be a non-capturing move segment.

Now, given a directed graph, construct a corresponding game configuration as follows. For each vertex, construct a merge which feeds into a split. Route the split outputs to the merge inputs of the vertex gadgets (merge + split) corresponding to the vertices the exiting edges connect to, using crossovers as necessary. Start the king on an extra input to any vertex (with a black piece to capture to let it enter the vertex).

Finally, equalize all of the "edge lengths" by adding extra black pieces along the output / input pathways as needed. If there are V vertices, and k black pieces along each edge, then the king can capture 2V + kV + 1 pieces if and only if there  is a Hamiltonian circuit of the corresponding graph. If the king has an alternative move available, capturing a simple chain of 2V + kV pieces, then determining whether that alternative move is legal is NP-complete.