Effect of increase in variance on covariance

Question

I have that the covariance between two random variables is negative: $cov(x,y)<0$. Suppose we replace $x$ by an rv, $x'$, that has higher variance but the same expectation (and, also, a negative covariance with $y$). I want to discern the effect this will have on $cov(x,y)$. In my understanding, $cov(x',y)$ should be less, in absolute value terms, than $cov(x,y)$ (and so less negative), because $x'$ is ‘more random’ than $x$, and should thereby have a weaker relationship with $y$. Is this correct? Thank you.