The first thing to remember is, since the ball is coming off the racquet with greater force, all things being equal, you can slow down your swing a bit and hit the same type of shot as before without any extra strain on your arm. This is especially important to keep in mind with any kind of awkward shot, like a wrist-flick to save a ball that's almost out of reach, or a desperation drive down the line while running wide. Don't hit these as hard as you can; chances are you're off-balance or not using ideal form anyway.
For normal rallying shots, what you need to keep in mind and practice is a return to good old "classic" form. With today's lightweight racquets, you might have adopted bad or sloppy habits like taking the ball late, hitting with all arm, or using excessive wrist. The heavier Babolat will make you pay for bad form -- the ball will jump off the racquet and zip through the air with a lot of spin, but all that extra force comes back to your arm. So remember what your earliest coaches told you:
- Step into the ball, so much of the force comes from the forward momentum of your body.
- Swing early and smoothly, don't wait until the last moment and make a rushed swing.
- Get the ball out in front, don't let it crowd you or get behind you.
- Get racquet speed by turning your hips and shoulders. I find it especially useful on the forehand to bring the left arm around as well as the right, and start the forehand by turning the shoulders so it feels like the left arm pulls the right along with it.
- Keep wrist motion to a minimum; get the racquet in position and on course early so you don't need any last-second adjustments with the wrist.