Carbs and the lack thereof don't cause colds. Viruses cause colds. Now possibly you're getting dry mouth from ketosis. Just keeping sipping liquids all day long or use some sugar free mints. Ketosis can be very dehydrating.

Other than that, you might be getting a cold or have a post-nasal drip or something. Sometimes my nose starts dripping at the back of my throat, sinus irritation cause by some unknown thing, and that can lead to a sore throat.