Unfortunately, I just purchased this product, and your question was posed over give months ago; thus, I have no idea whether or not you will read this answer. In any case, I will answer it in the hope that you will read it eventually.
My cats [all four of them :-) ] also catch the string, and the toy stops when they grasp the string firmly. There are two measures I take. First, I make certain that the removable "roller" (I do not remember the actual name of the small part at the bottom of the toy around which the elastic moves) is not full of dust, hair, or other debris. This is the only real maintenance the Fling-ama-String requires, but it must be done quite frequently. Second, I generally start my kitties with the slower speed. Almost inevitably they catch the string. Then I let them hold the string for a very short time (i.e., usually about 10-15 seconds--just long enough to give them the satisfaction of "catching their prey" and to wonder if this curious contraption will move again but not long enough for them to get bored and walk away, a time period that, I am sure, varies from cat to cat). After they have held it, I suddenly switch to the higher speed, which is usually enough to get it out of their grip. If I happen to begun on the higher speed, then I switch the toy off for again about 10-15 seconds and start it again. The speed at which I start it depends upon which of my cats is playing with it. Two are very skittish, so I start it on the slower speed for them, but for the other two "braver" cats, I begin the toy on the faster speed. This tends to keep their interest. Yes, I know this seems like a great deal of hassle for a toy that seems as if it should be one with which they should play independently, and it was a bit disappointing when I first used it, and I simply sat by the toy for about 20 minutes when I first usedit. Now, however, I rarely must do this since the cats have gotten used to it. Initially, all four were very apprehensive about the toy, and the "scaredy cats" were outrightly afraid of it. Specifically, the very movement of the string at the very beginning appeared to scare the cats, which is very strange given cats' natural instinct to track and to catch moving objects. Furthermore, the aspect of the toy that really frightened them for about a week was the fact that the string falls on their heads. However, with a bit of patience on my part and willingness to coax them gently to play with it, they have come to Fling-ama-String. I, too, enjoy the toy because of the sheer joy they get from it now.
Oh, I almost forgot: I have the sets of toys for my cats, and I rotate them such that they can play with only one set at a time. This keeps their interest inn all the toys.