If your camera supports it you can try registering faces to prioritize the auto focus for a few of the standout players.
If your camera doesn’t have any ibis and you’re shooting with a crop at 200mm I think maybe you’re right that 1600 isn’t quite fast enough. You could go higher or maybe try out a monopod to help stabilize.
There should be a tracking AF option where you can select a focus point and then let the camera track the point but your camera may not have that option either.
If no tracking AF try using a larger point. My camera calls it zone AF where it focuses on one of the 9 boxes made by the lines of thirds.
It could also be a “get good” situation where you just have to anticipate a little of what might happen and try to be ready for shots a few seconds ahead of time. Practice makes perfect after all.