"After they were cut, I was surprised to see how much water was coming out of the stump. They were wet. Though, I shouldn't have been surprised, because up in the tree where the rot had set in, it was wet there too. "
It shouldn't be surprising to see a freshly cut tree stump get wet. After all the tree draws water and nutrients from the ground through a system called xylem.
Trees use capillary action to move water from the ground, relying on the cohesion and adhesion properties of water molecules. When the tree is cut, this movement is suddenly halted. Cutting the tree releases the pressure that was keeping the water in the xylem, causing the water to seep out of the cut surface.
In addition to water, birches can ooze sap, a mixture of water, sugars, and nutrients, from the stump.
Trees hold a significant amount of moisture naturally, so even when they are not actively transporting water, the tree's cells contain moisture. This moisture is exposed to the surface when cut, making the stump wet.
If the tree is dead, it often dries up. These birches, judging by their stumps, looked very sick but not quite dead yet.