A couple of things:

Preparation of fire making materials is essential. Make sure you have plenty of tinder (wood shavings, dry leaves/pine needles, cattail fluff, etc.), kindling (small-medium twigs, split branches, etc.) and fuel supply (larger branches, limbs, split logs, etc.). Have everything ready ahead of your fire making attempts.

Wood will absorb more water/moisture from end grain then side grain. Breaking off and discarding the ends of small twigs and branches should, combined with removing the outer bark and outer layers of wood give you the best opportunity to obtain dry(er) wood. Look for dead twigs and braches still in the tree, these should provided better fuel then wood laying on the ground.

Pete