Every $\mathbb{Z}/6\mathbb{Z}$-module is projective

Let $M$ be a $\Bbb{Z}/6\Bbb{Z}$ module. We'll show that $M=2M\oplus 3M$.

If $m\in 2M\cap 3M$, then $m=2m'=3m''$ for some $m',m''\in M$, and this gives $$0=6m'=3m=9m''=3m''=m,$$ so $2M\cap 3M=0$.

On the other hand, $m=3m-2m$, so $2M+3M=M$.

Thus $M=2M\oplus 3M$. Note that $2M$ is a $\newcommand\ZZ{\Bbb{Z}}\ZZ/3\ZZ$ module and $3M$ is a $\ZZ/2\ZZ$ module.

Thus the claim you wanted to prove is true.

Then $2M$ is a direct sum of copies of $\ZZ/3\ZZ$ and $3M$ is a direct sum of copies of $\ZZ/2\ZZ$, and as you've noted $\ZZ/3\ZZ$ and $\ZZ/2\ZZ$ are both projective. Hence $M$ is a direct sum of projective modules and thus projective.