Prop icing can indeed happen, although it's highly unlikely to occur that rapidly (or catastrophically) unless the temperature is very low, air temp and dew point are less than 2 degrees apart, and humidity is above 80-85%.
I fly regularly in sub zero temperatures, usually 50-70% relative humidity and have never experienced prop icing. If you're concerned about it, a very quick and easy fix is to apply a very light spray of Canola based cooking spray ( Pam is one example) on the blade surfaces shortly before flight. It should be wiped off after you're done for the day...it's somewhat messy and will eventually become rancid if you don't
Yes, it sounds silly, and yes, it does work and does not affect flight stability or lift performance, nor does it get flung off by the spinning prop nearly as quickly as you may think.