Is drag higher when flying in a crosswind?
No, it's not true-- have you ever seen a birds feather's blowing sideways?

We fly INSIDE of the wind, not AGAINST the wind.  

You should draw the airplane symbol on the left-hand leg of the triangle, pointing in the same direction as the left-hand leg, and you should draw the "relative wind" arrow equal in magnitude and opposite in direction to the airspeed vector, i.e. the left-hand leg of the triangle.

Consider the fact that many of us have had the experience of actually BACKING UP in flight, relative to the ground.  Now, do you think the air was flowing backwards over the wings?