I can hear water running through the pipes day and night. It is not the immersion filling as I can tell the difference in the noise. The overflow is not running and I can find nothing to explain. When I shut off water supply the noise goes away and the meter stops.

Remove toilet tank lids and see if water is running.

Toilet is OK. Water is not running

If there is no toilet running-did you test with food color-, all faucets are off, TPV on hot water heater not leaking? Is this a slab house? If so, you may have a broken pipe under the slab.

I did not test with food color(how?), but all faucets are off and TPV on hot water heater is not leaking as well. Yes, it is a slab house. If it is a broken pipe under the slab how to find the place?

It is fixed now. The reason was the pipe to outside faucet had been broken.