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This is a technicality, but the wavelengths wouldn't actually change. They would be the same length from both perspectives. What actually changes between the two different observers is how long a foot/meter actually is.
Although even that isn't technically right either.
I think you might be incorrect. To a stationary person on the ground, the light they emit would be red or blue shifted depending on the direction they are going. Red or Blue shifting is the same as saying your wave length is changing.
What I think you meant to say is that the speed of light does not change at all. Which is correct. That has nothing to do with the wave length which can change in frequency.
Edit. Will clarify. Only the stationary person will see the wavelength or other term, frequency change. The person in the spaceship will not notice any changes in the light they emit as it will be entirely cancelled out by the time dilation.