Is there anything to try or does the fill valve just need to be replaced?
A toilet with a floatless fill valve isn't fully closing and is filling past the overflow tube (blue circle in image):
I have adjusted the adjustment knob (red circle) as much as possible to lower the water level. If observed closely in the image, the diagram at the right of the valve indicates to turn clockwise to "raise" or counterclockwise to "lower" the water level. Turning it completely counterclockwise essentially closes the valve and causes the water not to fill at all or to fill very very slowly, however it does not seem to adjust the water level as much as I need.
To my recollection, even when this valve was brand new it was difficult to get the adjustment right. It brought the water level to the edge of the overflow tube, but over time it has gotten higher, at first barely seeping over the top, and now it's overflowing constantly.
Another thought was to try to add some extension to the overflow tube but that might be more trouble than it's worth.
Best Answer
The internal gasket/diaphragm needs replacing, check out this link.
It does appear that you could extend the flush valve overflow tube as the height appears to be below the water level mark on your tank. That mark is actually the right level for optimal flush performance for your toilet. If the valve is worn out it may not help though, although the weight of the additional water may get it to stop. You might try opening the valve up and gently rinsing the gasket, in case there is debris or build-up interfering with proper operation (turn the water supply off first!)