regarding the headshake controversy on the 1100, I was one of those owners who had headshake occur, and as I recall it first showed up as the OEM front tire reached its end of life. A new tire reduced the effect, but it didn't go away completely.
At that time the tapered roller bearings seemed to be the consensus cure, and it worked 100% at curing mine. It was a bit of trial-and-error to get them adjusted just right, and the window for just right was extremely narrow, but it absolutely cured my issue. What didn't seem to get mentioned then, which I've now heard over the years, is that I probably could have cured the problem just by adjusting the stock steering stem bearings. The races definitely get pounded into the stem over time, and on a brand new bike they apparently do seat themselves a little tighter over the first year and probably need re-adjustment. The ST1100 factory manual suggests they be "inspected" at the 600 mile service, and every 8k miles. I also seem to remember that with the tapered bearings I had to re-adjust them once after 6-12 months because they had seated just a bit tighter as well.
If the front tire can be ruled out because its neither brand new, or heavily worn, I'd recommend that the OP at least attempt to adjust the steering stem bearings. Its not hard to do and there's nothing to lose. Mark the original orientation with a sharpie, and try tightening them a little bit at a time. I'd also add that grabbing the forks and shaking them or moving them side-to-side isn't sufficient to judge if they're not exactly right. If they're off substantially, then you will definitely feel it, but if they're close, but not perfect, you may not feel anything.
My experience is only on the 1100, not the 1300, so take that for what its worth, but I'd bet they're more similar than different.
The conclusion I and others seemed to find the most acceptable is that, modern performance motorcycles - and we placed the 1100 into this category as opposed to cruisers and full boat tourers - operate near the edge of stability in many situations. Decelerating through the 40-50mph speed with hands off the bars is one of those situations and revealed the compromise between rock solid stability and sport bike nimbleness that Honda made.
Having ridden about 200k miles on several sport bikes in the era before/after the ST1100, the ST1100 wasn't in the same universe as sport bikes of that era for handling. The phrase "sport bike nimbleness" does not apply to the ST1100 in any way, shape, or form. Also, I've owned too many bikes without ANY headshake, to know its not a normal or unavoidable handling trait, even on bikes with far steeper fork rake and much shorter wheelbase than the ST.