I like synthetic cork. The real stuff (Cinelli) is great and comfortable, but just not very durable. I have it on my fixed gear, but that only gets about 400-500 miles a year. Road bike has brand new SRAM synthetic cork on it. It is a good deal thicker and has a lot of spring to it, but so far amazingly comfy.

Make sure you watch some tutorials on how to wrap. Natural cork can tear if you pull it too tight, and some of the shifting you describe can come from wrapping too loose. The synthetic stuff benefits from a tighter pull when wrapping.