This is not true. Popa showed in "Orthogonal pairs of ∗-subalgebras in finite von Neumann algebras" (1983), that if $F$ is a free group with arbitrary cardinality than any abelian von Neumann subalgebra of the group von Neumann algebra $LF$ must have separable predual.
Edit: This doesn't even hold when $M$ is abelian since $\ell^\infty(\mathbb R)$ has no faithful state and hence does not embed into any $\sigma$-finite von Neumann algebra.
This is not true. Popa showed in "Orthogonal pairs of ∗-subalgebras in finite von Neumann algebras" (1983), that if $F$ is a free group with arbitrary cardinality than any abelian von Neumann subalgebra of the group von Neumann algebra $LF$ must have separable predual.