one week ago i bought a 1978 GL1000. I am a vintage honda tech and i know a lot of the common problems of old hondas, but ive never owned a goldwing.
so i did as much research as i could, and discovered that the shaking actually IS a common problem on older wings, and its MOST COMMONLY caused by
A. a difference in fork pressure (air assisted or otherwise) such as a blown seal, or mis matched air pressure, which can cause the load to vary quickly and the suspension pogos and its turned into a wobbly motion.
B. steering head bearings - after so long, the grease dries up and you get metal to metal contact from the needle bearings to the race. might not be noticable during normal operation, but the very tiny flat spots that are now in the face of the needle bearings will cause oscillation at low to moderate speeds.
C. a heavily bent rim/ loose spokes (spoked or comstar wheels) or worn wheel bearings. This is least likely becuase it would be most noticable, but definately check this out anyway
I've seen "B" as the most common cause, especially on my own 1980 CB900C.
you probably have a slight amount of BOTH going on for it to be enough for you to dump the bike.
one way to tell is to look at your tire itself, is it cupping? that is an indicator that it is your forks that need to be looked at. and although it COULD be wheel bearings or a bent rim at this point, if the wheel rolls smooth and you cant see any wobble from your vantage point on the seat while riding, than your probably safe and youd best start at the front suspension
if you dont have another goldwing to compare, you might not notice the steering becoming sluggish. but i would have the bearings replaced as a precautionary measure.
you have to understand that these bikes were not designed to wobble when new. so something, somewhere, is wrong, in order to cause this.
I hope I have given you a good jumping off point to look for problems.