This is the whole point of poker. You're probably playing differently when your own money is on the line vs when it's a freeroll. That's the whole point. Poker is supposed to be played for money.
Proper bankroll management should mitigate that scared feeling you get when your money is on the line. For tourney players you should have 100 buy-ins for mtts and 50 buyins for stts. If you can stick with that then you shouldn't feel scared to lose a buy-in and be able to play your best game. Additionally, your risk of ruin, or chance of busting your entire roll, is greatly reduced with this method. However, this only works if you're a winning player. If your a losing player, NO bankroll management will work since you will end up broke in the long run no matter what. Hopefully by the time that happens, you've played enough to be able to find the leaks in your game and fix them.
Good luck!