A coupled-channel approach is applied to the charged tetraquark state $T_{cc}^+$ with special attention paid to the three-body dynamics. The three-body unitarity is preserved as both the pion exchange between the $D$ and $D^*$ mesons and the finite $D^*$ width are taken into account simultaneously. The low-energy scattering parameters, namely the scattering length and effective range are extracted with a low-energy expansion of the $D^*D$ scattering amplitude. The compositeness parameter is calculated and is found to be close to unity, which implies a molecular nature of the $T_{cc}^+$. By making use of the heavy-quark spin symmetry, an isoscalar $D^*D^*$ molecular partner of the $T_{cc}^+$ with $J^P=1^+$ is predicted under the assumption that the $D^*D$-$D^*D^*$ coupled-channel effects can be neglected.