Another answer noted that tread pattern and tire pressure make a larger difference than the width. If you have a smooth tread pattern at high pressure, you will roll more easily and the tire won't "squirm" going around corners. Tire compound can also affect handling. The worst tires I ever rode on were a cheap narrow-ish (maybe 28c) set with raised tread made of a harder rubber compound. The traction was terrible: I could lock the rear tire even under fairly light breaking.
There is a connection between width and tire pressure. Generally narrower tires support higher pressures, and also require higher pressures to support the same weight on a smaller volume of air. Up to a point, running higher pressure will reduce rolling resistance. If you run the pressure too high it may actually slow you down as small bumps will jostle you more. You can try running wider tires at their max pressure to reduce rolling resistance and see how they feel, but it will likely give you a rough ride and have fairly poor contact with the road surface, compared with a narrower tire at the same pressure.
The 32C tires will likely give a noticeable difference: the ride may be a little rougher but rolling resistance should be reduced, and you'll probably feel a little more certain on corners. Contrary to the other poster, rolling resistance is less than air resistance, but is a significant factor at moderate speed: if you're riding at 25MPH, air resistance totally dominates, but at 15MPH, rolling resistance is non-negligible.
I would say just get the 32C tires and start riding on them: it's probably not worth the wait. There's also a good chance that 28C might be too much of a good thing.