In general with your assumption

$$Ric(\omega)=\sqrt{-1}\partial_i\bar\partial_j\log\det g_{i\bar j}$$

so by this formula we have always $Ric(cg)=Ric(g)$, and if $h,h'$ satisfies in $Ric(h)=−g$, $Ric
c(h')=−g$ then in general it is sufficient to use of this formula and remove $\partial\bar\partial$ of bothsides and we get $\omega^n=e^c\omega'^n$ or $\det h_{i\bar j}=e^c\det h'_{i\bar j}$ for a constant $c$, so a lot of such metrics exists

and by Aubin's proof (he was a first French Mathematician who proved it and later Yau gave different proof) and Calabi himself also gave a proof for unicity of such metrics when the first Chern class is negative

if the kahler metric $h'$ lies in a $ c_1(K_X)$,then $h=h'$. See Tian's book

In fact if $\omega'$ be the corresponding metric of $h'$ then by $dd^c$-lemma we can write $\omega'=\omega+\sqrt{-1}\partial\bar\partial \varphi$ and so we will have a non-linear parabolic monge-Ampere equation
$$\frac {(\omega+\sqrt{-1}\partial\bar\partial \varphi)^n}{\omega^n}=e^c$$

which its solution is unique. See theorem 19.1 page 129 Lectures on Kahler geometry Andrei Moroianu