I'm far from an expert on these things, but my Bianchi Sport SX also gets wiggly when I take my hands off the bars. It was a bit unstable for the previous owner, @
balindamood, as well. Interestingly, the only hardware in common for our riding is the frame (excepting the fork), and I've used the string test to make sure it's pretty well aligned.
What does seem to be in common, is having weight too far back, and not enough of it on the front wheel. Brian used a rear rack and trunk bag, I had it set up in sort of a French/Eddy fit with knees behind KOPS and an 80mm stem. The new fork has more rake for less trail -- I'm not sure if that helped or hurt the stability, but it makes the bike easier to control with a light touch, so I'm content. I've scooched my saddle forward a bit and am pretty much at KOPS now, considering a 90mm stem since I can now reach forward a bit more. Moving my weight forward seems to have helped. The bike still gets shaky if I take my hands off and pedal at 20+ MPH, but now I can ride no-hands smoothly at 15-18 MPH.
I think the ultimate solution might be to find a frame that's 10mm shorter in the top tube, and make up for it at the stem so that a little more of the front wheel is under me. (My Bianchi Eros, one size smaller, doesn't wiggle.) Perhaps longer chainstays would help, too. Might be looking at a custom frame if too many of these odd demands add up.
That's a lot of blathering to say moving your bars up and back may have indeed hurt your cause. One thing to try is to pull your rear wheel back in the dropouts -- I like to line up the rear axle with the seat stay, and can see that your wheel is ahead of that.