Yes, moving water can evaporate faster than still water. When water moves, the molecules rub against each other and this will make the water warmer over time. The higher temperature will make the water evaporate more quickly. The difference, though, is small, especially in the ocean where other parts of the environment play a bigger role. The physical process that heats the water also slows it down, which means something has to keep the water moving. You can see this if you stir the water in a sink or bathtub. If you get the water moving and then leave it alone, eventually it will stop moving. In the case of the ocean, the currents are being "stirred" constantly by the winds.

I don't think that moving water evaporates (much) faster than still water but air blowing on water will make it evaporate faster. This is how a wind chill works. What you noticed is that heat will conduct pretty rapidly through water via water molecules moving around (also known as diffusion).