60mm is a huge jump from 110. Even if going this route to solve the problem I would suggest a lower delta on the first iteration or two.
Aside from trail, fidgety steering is also a function of how one's weight is distributed over the steering axis, relative to ones "ideal" position. A shorter or longer stem doesn't in itself guarantee anything - a stem that is too short relative to ones ideal overall reach will quicken steering, and subsequently a stem that is on the longer side relative to ones appropriate reach may slow things down. Quick (and maybe oversimplified) example: My previous frame had a 54.5 top tube and 90mm stem, my current frame has 53.5 top tube and a 100mm stem. Both frames have identical rake and trail. Both bikes handle pretty much identically despite the different stem lengths... my weight is positioned similarly relative to the steering axis despite the different sizes/stem lengths. If I were to have run a 100/110 on the 54.5, or a 110/120 on the 53.5, I might expect the steering response to slow as my stem length relative to my ideal reach would have increased. And visa versa going short.
A non cycling example would be to hold a bowling ball close to your stomach, then hold it a little further out. You can rotate at the waist and still have a good degree of control without it's mass becoming unwieldy. When going too far out from the center however, you'll notice less control. My size/stem example above puts the "bowling ball" in the roughly the same place relative to myself despite the different measurements.
This conjecture btw is from my own experience on having worked with a number of ill-fitting frames, and is somewhat subjective so take with a grain of salt.
But as Bob mentioned the weight distribution over a frame that is incorrectly sized may (or may not) affect things.
The longer inseam would suggest a shorter top tube, but I'm gathering an exchange of frame size is not an option. Maybe the dealer would consider a restocking fee? Overall a small price to pay considering the original good deal.
That said, I have gotten both smaller and larger frames to work using various stem lengths, handlebar drops, headset spacers etc. but always better to start from the closest to ideal size.