Starfleet sensor technology, which can measure quantum-level disturbances in matter, is incapable of detecting Changelings who have shape-shifted into another person or object. Additionally, Changeling can imitate the properties of metal, cloth, living tissue, and even fire and fog. They can even produce functioning electrical components such as combadges. All of this suggests that their molecules and even their atomic and subatomic composition can change along with their shape. Therefore they are neither consistently polar or nonpolar.
As for their natural "orange goo" state, very little is known of its composition, although it is known to contain "morphogenic enzymes." Enzymes are catalytic proteins, so they do contain at least some organic molecules. In human biology, proteins usually have both polar and nonpolar domains to coordinate their chemistry.
As for the issue of mixing, I cannot think of any occasions on which a Changeling mixed with any other form of matter while in their natural state. Indeed, they seem to have a remarkable ability to remain intact even when in contact with materials which would usually disrupt or absorb a liquid. For example, Odo doesn't even get Lwaxana Troi's clothes wet when he's forced to, ah, melt with her in DS9: The Forsaken. The aforementioned case of Laas shifting into fog does give an example of a Changeling mixing with air, but since gases always mix regardless of polarity, this doesn't tell us anything.