Here's a counterexample:

Take $K=\mathbb{R}$, $L=\mathbb{C}$, $A=\mathbb{R}[x,y]/(x^2+y^2-1)$. Then $A$ is a Dedekind domain with class group cyclic of order 2, and $A'=A\otimes\mathbb{C}$ is a PID. We can take $M$ and $N$ to be non-isomorphic projective rank 1 modules over $A$, which both necessarily become free after tensoring with $\mathbb{C}$.

Explicitly, we can take $M=A$, $N=(x,y-1)\subset A$.

Here's a counterexample to the same statement for infinite dimensional algebras:

Take $K=\mathbb{R}$, $L=\mathbb{C}$, $A=\mathbb{R}[x,y]/(x^2+y^2-1)$. Then $A$ is a Dedekind domain with class group cyclic of order 2, and $A'=A\otimes\mathbb{C}$ is a PID. We can take $M$ and $N$ to be non-isomorphic projective rank 1 modules over $A$, which both necessarily become free after tensoring with $\mathbb{C}$.

Explicitly, we can take $M=A$, $N=(x,y-1)\subset A$.