As briefly mentioned in post #3, and I have brought up a time or two in various threads, I believe a rear riding height adjustment can make a difference in handling and high-speed wobble. Raising the rear (or reducing carried weight) makes the forks more vertical, which quickens steering and reduces stability, while lowering the rear has the opposite effects.
When I got my '01 ST1100 last August, I thought it handled like a hippo, especially compared with my '96 Nighthawk 750, even though it's fully dressed. The previous owner is lighter than I am, and having read about the above-mentioned steering geometry effects, I decided to raise the rear, by about 1/2" at the shock; not sure about the actual ride height.
The bike now handles much better, requires less counter-steering effort and feels lighter, more controllable, and predictable. If I over-do the adjustment, I believe I could get to the point that it would wobble at certain speeds with certain loading. I think a lowering of the rear of a wobbly 1300 will reduce the tendency to wiggle, all other things being equal.
The 1300 has the reputation of handling better than the 1100, and to suffer the wobble problem more. Coincidence, or a combination of design and set-up geometries? Either bike can have the steering geometry adjusted by rear riding-height changes, and for much less cost than new tires, fork braces, steering dampeners, steering-neck bearings, etc.
If I had a 1300, I'd experiment as I have on the 1100. In fact, I'm thinking of raising the rear a tad more to see. I have hit one end of the handlebar at various speeds with my hands off, and the one reaction shake immediately stops, so I'm not near the verge of instability. I encourage someone try this and report back with your findings. Anyone? Frye? Bueller?