SM, I would guess that you have a bad thermocouple. These check to see that the pilot is actually lit before the main gas supply valve opens. They are not hard to replace but sometimes tricky to position properly. Parts can sometimes be a problem on the weekend.
TT, you are probably on the right track.
I recognize that there is a cost but I would call a pro in both cases. Your cold weather putting more usage on your furnaces might be part of the problem. And the weather is probably putting more stress on your repair people as well. Get on the phone now.