Conflict with the properties of convolution
I believe that the problem you have with this exercise is mainly caused by bad notation. Writing convolution as

$$z[n]=x[n]\star y[n]\tag{1}$$

(which I think is bad notation) it is tempting to believe that

$$z[n-1]\stackrel{?}{=}x[n-1]\star y[n-1]\tag{2}$$

which is not the case.

If you take the definition of (discrete) convolution

$$z[n]=\sum_kx[k]y[n-k]\tag{3}$$

then it's obvious that

$$z[n-1]=\sum_kx[k]y[n-1-k]=\sum_kx[n-1-k]y[k]\tag{4}$$

i.e., only one of the two signals needs to be delayed by one sample, not both.

A better notation is

$$z[n]=(x\star y)[n]\tag{5}$$

Now you can define $x_1[n]=x[n-1]$ and $y_1[n]=y[n-1]$ and you have

$$z[n-1]=(x\star y_1)[n]=(x_1\star y)[n]\tag{5}$$

If you view $z[n]$ as the output of a system with impulse response $y[n]$ and input $x[n]$, then it should also be obvious that delaying $z[n]$ by one sample is equivalently achieved either by delaying the input signal, or by adding a delay to the impulse response, but not by doing both.