Your question ("is $M$ homeomorphic to $T_f$?") is answered in the affirmative by Theorem 2 of Stallings' paper *On fibering certain 3-manifolds". You will also need his Theorem 1. Here are the statements (slightly simplified).
Theorem 1: Suppose that $M$ is a compact connected three-manifold. Suppose that $\Gamma$ is a finitely generated normal subgroup of $\pi_1(M)$ whose quotient group is $\mathbb{Z}$. Then there is a surface $F$ properly embedded in $M$ so that $\Gamma = \pi_1(F)$.
Theorem 2: With hypotheses as in Theorem 1. Suppose that $M$ is irreducible. Suppose that $\Gamma$ is not $\mathbb{Z}/2\mathbb{Z}$. Then $M$ is a surface bundle over the circle, with $F$ isotopic to a fibre.
Also, Stallings does not cite Waldhausen. Consulting my memory, this is because once you have $F$ in your hands (and all the group theory hypotheses), it is "easy" to show that $M - F$ is homeomorphic to a product.