I'm assuming $\Lambda=\mathbf{Z}_p[[\mathbf{Z}_p]]\cong\mathbf{Z}_p[[T]]$ (with a topological generator $\gamma$ of $\Gamma=\mathbf{Z}_p$ going to $1+T$).
Let $Z$ be the $\mathbf{Z}_p$-torsion submodule of the $\Lambda$-submodule $X^\prime=\bigcup_{n\geq 0}X^{\Gamma^{p^n}}$. Since $X^\prime$ is finitely generated over $\Lambda$, $X^\prime=X^{\Gamma^{p^n}}$ for some $n\geq 0$. This means that $X^\prime$ is annihilated by $(1+T)^{p^n}-1$, and hence is a finitely generated module over $\Lambda/((1+T)^{p^n}-1)$, a finitely generated $\mathbf{Z}_p$-module. So $X^\prime$ is finitely generated over $\mathbf{Z}_p$, and thus its $\mathbf{Z}_p$-torsion submodule $Z$ is finite. Conversely, if $Z^\prime$ is any finite $\Lambda$-submodule of $X$, then $\Gamma$ must act discretely on $Z^\prime$, so $Z^\prime$ is fixed element-wise by some open subgroup of $\Gamma$, i.e., $Z^\prime\subseteq X^\prime$. Since $Z^\prime$ is of finite cardinality, in fact $Z^\prime\subseteq Z$. Thus $Z$ is the unique maximal finite $\Lambda$-submodule of $X$.
Now let $Y$ be the $\mathbf{Z}_p$-torsion submodule of $X$, i.e., $Y=\bigcup_{n\geq 1}X[p^n]$. This is a $\Lambda$-submodule, hence finitely generated, and it follows that, for some sufficiently large $n$, $Y=X[p^n]$, so $Y$ has finite exponent. Because $X$ is torsion, the structure theory for finitely generated $\Lambda$-modules shows that $X/Y$ is finitely generated over $\mathbf{Z}_p$, and since it is torsion-free by construction, it is finite free over $\mathbf{Z}_p$. Conversely, if $Y^\prime\subseteq X$ is a $\Lambda$-submodule killed by some power of $p$, then $Y^\prime\subseteq Y$, and if $X/Y^\prime$ is torsion-free over $\mathbf{Z}_p$, we must have $Y\subseteq Y^\prime$, whence the equality. So $Y$ is unique.
