Sometimes things sound great when playing on one string, but as soon as you play the same material on another string, the tone disintegrates into a scrape. Typically this is because your bowing arm height is changing too quickly, and the bow is being pushed into the new string before it has time to settle into its stroke. Rather than forcing the whole arm abruptly upward or downward, use your elbow as a hinge to allow the bow to slide laterally without changing the bow’s contact point.
Listen carefully to the very first instant that the bow strikes the new string. If it crunches for the first split second before clearing up, then the bow has arrived on the new string with too much pressure. Start the stroke with a bit less pressure, and gradually increase to a full sound once the string has been set into motion. Practicing very slow string crossings on open strings will allow you to hear whether the bow is coming down heavily on the new string. You want to place your foot on the ice carefully, rather than jumping onto it! It’s a tentative sort of motion.
Another common error is to arc the bow as you cross to a new string. This means that for an instant, the bow is no longer parallel to the bridge, and the tone will sound mushy and uncertain. Practice your string crossings in front of a mirror, and observe whether the bow travels in a straight line even though the height of your arm is changing. If the tip of the bow is dipping or wobbling, slow down the stroke and exaggerate a straight line until that motion becomes the norm. The direction of the bow is just as critical as its pressure!
You can speed the process along with a short daily exercise. For a few minutes, play long strokes on two adjacent open strings, alternating between them. Start with long strokes and gradually shorten them. Keep the tempo slow enough that your ear can monitor each shift. When the tone is fairly clear, add a simple left hand pattern to your crossing strokes, but don’t change the way you are crossing strings. This will help your body understand that playing on a different string doesn’t mean you play differently, just that the angle of the bow changes a bit.
If you find that your string crossings deteriorate at higher speeds, go back to the slow motion crossings for a minute or two to get things working again. Smooth string crossings are largely a matter of timing between the motion of your arm and the stroke of your bow. It has nothing to do with applying more force! As your control of this motion improves, you will find that the bow seems to disappear as it crosses from string to string, and that your tone quality is the same no matter where you are on the fingerboard.