I’ve worked on my Campbell hydrant way too much and have replaced just about every part on it.
First thing I’d try is to adjust the handle position on the rod that goes into the standpipe to push rod farther down when handle closes.
This probably won’t work if the plunger at the bottom of the standpipe needs replacement.
So.. 2nd thing is to disconnect the push rod from the handle and unscrew the hydrant head from the standpipe. This might be difficult for you because the wood post is the way. Then pull up the pushrod that’s internal to the standpipe and replace the plunger attached to the bottom of it.
If the plunger doesn’t shut the water off completely, as is happening now, you can still be leaking water and never know if the water drains out the port at the bottom underground instead of making its way up and out the standpipe/hydrant.
The only way I can tell mine is leaking is I installed a pressure gauge in the house downstream of the valve in the house that supplies the hydrant. Only by turning off the supply valve and watching the trapped water in the line to the hydrant lose pressure do I know the hydrant is leaking.