Eastlery winds DECREASE with Altitude and become Westerly as you get higher in the Northern Hempisphere (Earth Rotation).
Only Westerlies become stronger with Altitude.
This is not correct. Of course you can find specific weather situations where the wind behaves as you say, but in general there is no such correlation.
Addendum: Upper winds are generally westerly (in both hemispheres), but that doesn't mean that an easterly surface wind will generally turn westerly with increasing altitude -- certainly not at 6000 ft.