Like Andrew, I suspect that you just have a valve turned wrong. Look on the underside of the hatch over the valves and there should be a diagram showing the correct orientation. If not, then your owners’ manual should have one. If your pump is running constantly then it’s likely that it’s the fill valve on the lower right that’s wrong, and the pump is just cycling water back into the tank.
If that doesn’t do the trick, then check the filter on the pump as it may be clogged. It’s the clear plastic container connected to the pump.
Another unlikely cause could be a broken valve. And always a chance that there’s not actually water in the tank. 😝
It’s no surprise that your tank drain just dripped - that seems to be a common problem that has been discussed here more than once. One good solution is to do just as you thought and just run the water into the grey tank.