What's weird about that? My armpits could be sweating (e.g., wearning a t-shirt + sweater) but my hands (no gloves) could be freezing.

Different parts of your body can feel hotter or colder, because the sensation of temperature is not about your internal body temp, but the heat transfer at the skin.

Hands with a pair of gloves on will feel colder than e.g. armpits under 3 layers of clothing. It's absolutely possible to sweat and still feel somewhat cold.

Almost. I typically sweat due to too much clothing, which results in me feeling cold.