Without seeing a photo, it sounds like you are using the wrong AF settings. As stated above for Nikon use AF-C (AI servo on Canon). You are fine in Aperture priority, If you're comfortable shooting there stay there and don't worry about manual mode. (don't have heart attack manual guys lol) Use a single focus point and aim straight for the players face. If you let the camera pick the focal point you'll miss a ton of good shots.
Take your ISO off of auto. If it's outdoor lacrosse you should be fine with anything below 400. In good light go as low as the camera will set it and still get a good shutter speed. You shouldn't have to limit your shutter speed to 1/1000 or higher either. F2.8 on a sunny day will definitely give you some fast shutter speeds. Sometimes I like to stop down around F4 to F8 to get multiple players sharp like when an attackman is driving through a defender at the crease.
Good luck and keep practicing. I look forward to seeing some of your photos.