The internal structure of the near-threshold exotic hadrons, $T_{cc}$ and $X(3872)$, are studied by respecting the decay and coupled-channel contributions. The effective field theory model is introduced to calculate the compositeness, the probability of finding the hadronic molecular component. Applying the new interpretation scheme for the complex compositeness of unstable states and using the bare energies of the compact component estimated by the constituent quark model, we find that approximately 50 \% of the $T_{cc}$ wavefunction is occupied by the $D^{0}D^{*+}$ molecular component. This indicates that the structure of $T_{cc}$ is primarily dominated by the $D^0D^{*+}$ component, having the nearest threshold. However, other components, such as the isospin partner $D^{*0}D^{+}$ channels, also provide non-negligible contributions. On the other hand, about 90 \% of $X(3872)$ is composed of the $D^{0}\bar{D}^{0*}$ molecular component. This is because the non-composite bare state and the charged $D\bar{D}^{*}$ threshold are located relatively far from the $X(3872)$ and contribute very little to its internal structure.