Yes there is a seacock and I close it at the dock but not with the engine off when I am sailing. Why would be be dry? All the lines are below water until the exhaust. The boat is new to me and the strainer has just moved to the top of the priority list.
Not being able to see your installation it was just a thought.. On my own boat also with a raw water Yanmar I always close the seacock whenever the engine is not running.
Yes, mine will also refill the strainer with the seacock open and the boat level but on the times I have been sailing and not closed the seacock I have sometimes observed the level in the strainer going down.. I'm thinking this may be caused by the boat heeling or even water passing over the open intake and drawing out by some low pressure effect.
Try closing it as soon as you shut down the engine. It's an easy test and if it makes no difference then you can start looking elsewhere.