The effect is pretty easy to envision, really.
You have a sample cup filled with oil. Good oil is almost a perfect insulator, so in the sample cup you have a capacitance (essentially fixed: cup geometry and the dielectric constant of oil) so you have much more capacitive current than resistive, giving a LOW (very low) power factor, because you have a high resistance (real power loss) paralleling a very small capacitance.
It the oil is deteriorated, the resistive component changes due to polarized components in the oil, therefore the resistive component is higher. In any reasonable oil sample, there will be little change in the capacitive element, but with "bad" oil, the resistance goes down, the watts loss goes up, but the capacitance stays more or less the same. Therefore, the power factor INCREASES.
These changes are very small. Doble's literature says that oil with a power factor of 0.3% or less is acceptable for service, 0.5% is considered "doubtful" and should be tested by other means, and 1.0% should be investigated, reconditioned or replaced. Of course, the Doble oil power factor test setup is a field test and not as rigorously standardized as a laboratory test.
As to the change in your transformer, remember that the affinity for oil in the transformer is shared between cellulosic components, the paper and wood used in insulation and support, and the oil itself. Hot oil will absorb oil from "wet" cellulose. As the oil cools, the moisture will migrate back to the cellulose, as long as there is enough time for this to take place. If the change is rapid, the water in the oil may precipitate out of the oil, resulting in visible moisture in the tank.
It's all not very static, and that makes it interesting to interpret.
old field guy