Many times I've faced this difficulty, when I play against higher rated players , I automatically go for a draw and not a win. What happens is people are much more afraid of a loss than a draw. I have also drawn against 1600's and 1700's in a winning position.
Why this happens is the lower rated players think that their opponents may outplay them in their endgame, or even they themselves might make a foolish blunder.
The way to get out of this is to analyze your current position- or after the game you'll end up telling everyone " I was winning but I was not confident, so I asked for a draw." , After your analysis look if there is a possible win. If you're not sure, play a few moves calmly but a little cautiously , If you see an advantage, GO FOR IT. Do NOT doubt yourself . And if even this is not working think that chess is only a game. Losing will dissapoint you, but you will learn a lesson and you will not make the same mistake twice.
P.s- Learn from other's mistakes rather than yours.