Why does the toilet keep running?
Have you verified that the flapper valve is seating properly? Even a brand new toilet can run if the flapper valve gets hung-up and is not allowed to close fully.

If you know for sure that the flapper valve is moving freely and tries to seal the drain pipe then try turning off the water flow and see if water is flowing in from the septic system. That's the only way I know that sediment would be able to enter the tank unless it was already in the line somehow.

Check those and report back if neither help to further evaluate.