With front squats, remember 3 important things regarding technique: elbows up, chest up, and head up.
If the movement bothers your shoulders, try a modified grip. Put the heels of your hands together, just like you were doing a hands together curl. With this hand spacing, grip the center of the bar, and step under, so that the bar is resting high on your front delts. It will look like you have just finished a hands together curl, at the top of the movement.
Many Olympic Weightlifters use this modified grip on front squats to save their wrists, elbows, and shoulders, from any more stress, from all of the heavy cleaning that they do.
If front squats are bothering your back, you are going into the squat knees first, not butt first, which is exactly backwards. Always bend at the hip joint first, stick you butt out first, and keep as much of the movement in the big muscles of your gluteus, with your balance on your heels.
Most problems in lifting can be solved by either pulling harder, or pushing harder.