The Sony P93 takes sharp photos and it may be hard to find a replacement. Its photos are sharper than those from most of the new ultra thin cameras.

As far as shutter lag, the Sony P93 has a shutter lag (0.4 second), that is better than some new cameras.

The current small Sony cameras have a shutter lag of 0.3 second.

Your best bet is to use the two step method of taking a photo:
Press the shutter button half-way down.
This gives the lens time to focus.
Now reframe the shot and when the action occurs, press the button the rest of the way down.
By using this method, you get the shutter lag down to 0.1 second.

If you are shooting outside with good lighting, you should get good action shots.

If you are shooting action photos under artificial light, you are likely experiencing blurred images, either the whole photo or just blurred legs and arms of people.

This is caused by not having enough light and the camera must select a shutter speed that is too slow.

A new camera will not fix that problem,
unless you are willing to spend over $750.

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