Water at spigots, shower, toilet, etc. with pump shut off?

If the tank overflow line is plugged then there could be pressure in the tank. Air in the hot water tank can also cause a limited flow as can residual pressure in the water lines.

Maybe some more information is needed. How long do you continue to get water? I get water out of my spigots with everything turned off for quite a while and more pressure than I would expect. Even with the batts off so I know the pump isn't running. I think 2 things are at play; the system maintains pressure until all the water is emptied from the lines, and as things heat up during the summer - especially if you had your water heater on, pressure builds long after you shut the pump and WH off. I'll go to my camper in storage and despite emptying my lines when I put it away, I still get a little water to come out to wash my hands. Infrequently the power will go out at the house and I'll have water for quite a while until all the pressure is relieved. Same principal.

If however you think you have evacuated the lines and you still get constant pressure, yeah, your pump is still on. I think you would hear that running though wouldn't you?

Some rigs have an accumulator tank which would hold the water pressure a little longer but it will still stop after a little bit

MFL pretty much said it all. you,ll get a bit of a drips but not running .

Seems obvious that the pump is not truly switched off. Unless the pump is running the water cannot continue to flow, other than a bit of residual pressure allowing slight water flow for a bit.

I suspect your on/off switch has failed. Most pumps make a little noise when operating. Some are fairly quiet, but you should hear it running.

Jerry