You shouldn't move your seat to compensate for reach unless you have a frame that is absolutely too big, which it doesn't look like you do. The bars through the axle while on the hoods deal is a just a really quick measurement of stem length. It may not work for everyone.

I had a similar problem and went from a 120mm to a 90mm. The 90mm fits very nicely. I still get a good aggressive position (9.5cm drop from the seat to bars) and reach without any discomfort.