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I agree with you. Water facilitates heat transfer, and the surface of skin is about 95f, so it seems that below that temperature, and assuming you aren't already sweating for the evaporative cooling effect, humid air should move heat out of your body more effectively than dry air.

I think “dry” and “damp” are two extremes. I feel warmer in 45% rH at 67°F in winter than at 30% rH same temperature, but the air doesn’t feel damp (OTOH 60% would start to approach feeling damp).

That being said, I think at normal conditioned air temperature ranges more humidity virtually always feels warmer. I’m not exactly sure where the cutoff is, but I’d guess below 60°F might be a decent guess?

I think you're right and actually the extra moisture in the air would also mean that raising the temperature of the house requires more energy input since the specific heat capacity of water is higher than air.