Your sinuses humidify the air you breath in,so they become dry-the water evaporates-mucus becomes thick-you get drying and a little pain a little bleeding.

Is it worse during cooler weather-when there is less water in the air( relative humidity isn't the problem-actual water content is the problem) ?
Or is it worse in warmer months when there are more allergens in the air?

If your problem isn't allergic-and it is just dehydrating sinus and nasal membranes-
then sometimes it helps to loosely cover your mouth and nose-so the air you inhale is mixed with the higher water content air you exhaled-granted it would be a tiny tiny difference in inhaled water content-but occasionally a tiny difference is enough.

PS Occasionally some medications can make these problems(drying) worse.See what your MD has to offer.