A lower coupon bond exhibits higher duration, which means higher price volatility with changing YTM.

A lower coupon bond also exhibits higher convexity. However, with higher convexity, bond prices rise more and fall less.

So, a low coupon bond has higher duration and higher convexity, yet has higher and lower price volatility at the same time?

Is someone able to help with this contradiction?

The change of the price $P(y)$ if the yield changes from $y$ to $y+\Delta y$ is $$\frac{P(y+\Delta y) - P(y)}{P(y)} = - D \Delta y + \frac12 C \Delta y^2,$$ where $D$ is the duration and $C$ is convexity. For small $\Delta y$ the square is much smaller. Thus the duration component dominates.