My experience is very similar.
I had very old Cypress knees in my tank and used a bag with heavy stones in it and it still didn't get water logged. Mine soaked in a kid pool for a couple months before.
So... my ideas were,
Use of course the Aquarium safe kind, silicone to affix it to a paver stone or slate. It actually sat on some paver stones to get the height I wanted and so the fish could swim under it and hide.
And you can also drill a hole in slate and use large stainless steel screws and probably silicone and drill / put the screw in the wood with the silicone too and it might take some time to cure.
With mine it seemed the wood structure inside might not hold a screw drilled in since it seemed hollowish or probably just really old, and might pull out easily.
My other idea was zip tie a bag of sand in a couple spots. I used a sand bag - bag to put the rock in and zip tied it shut. Inside were a couple heavy kinda flat river stones seen in the early post pics.
It seemed to me these would help and in a matter of time the CK wood, would soak up the water enough. But mine did not and wanted to bob and flip over sideways. Even after a year.
You can see the CK wood in post #4
FW & SW
Favorite tank right now - 72G Bowfront Wadai Tetra, Cardinal Tetras, Ghost Glass Cat Kryptopterus bicirrhis and Clown Loach Cory haven
New to FW fish? First tank in a long time or EVER??? Need some info on keeping fish?
Start here - Aquarium Advice Article