The light is really working against you, as posted above. One alternative is to go with the type of shot you almost got, by panning. That is, following one player with the lens so that his motion is cancelled out by the following. Everything else will have intentional motion blur. I don't think any picture mode will help with panning. Here is an example I took recently:
I was using a 120mm manual focus lens for this. I focused on the road in advance. Panning works with a slow shutter speed. This shot used 1/20 sec., which allowed me to stop down a lot, f13 at ISO 100. It looks like she's going fast but it's only 15mph.
For your shot, you could focus in advance on a point on the field where you expect a player to cross at speed. Then turn off the autofocus. Set the camera to M mode, shutter speed to 1/30 or so, and take a shot of that spot. That should allow you to adjust aperture to get enough depth of field and ISO to get a good exposure. (The exposure shouldn't be that much different when the player is in the shot, unless the uniforms are all white.) Now you have to figure out whether that shutter speed is too fast or too slow with some more practice shots. The ideal speed is related to how well you can pan along with him, his speed, your distance and focal length.
I used a K-7 for my shot which has two burst modes. The highest speed setting is 5 frames a second, really too fast for this. There's a slower setting that your K-x should be able to match, and I used that for my shot. Then I could press the shutter button in advance, follow the action, and have several frames to choose from. Another alternative is to just take one shot when the player hits the right point.