We investigate the internal structure of two exotic tetraquark candidates, ${\chi }_{c1}(3872)$ and $T_{cc}^{+}(3875)$, whose masses lie within $1\MeV$ of the $D^0{\bar{D}}^{*0}$ and $D^0{D}^{*+}$ thresholds, respectively. Using the effective-range expansion (ERE) supplemented by resonance compositeness relations on the second Riemann sheet, we extract scattering lengths $a$ and effective ranges $r$ directly from the experimental masses and widths, and quantify the $D{D}^{*}$ molecular content through partial compositeness coefficients $X_j$. A two-channel framework couples each state’s inelastic detection channel ($\Jpsi\,\pi\pi$; $D^0{D}^0{\pi }^{+}$) to the near-threshold molecular channel. Full Monte Carlo uncertainty propagation with $N=50000$ Gaussian samples yields asymmetric $68\%$ confidence intervals for all observables. We find large negative scattering lengths $a=-{8.49}_{-1.05}^{+0.93}$ fm for ${\chi }_{c1}(3872)$ and $a=-{14.57}_{-1.66}^{+1.70}$ fm for $T_{cc}^{+}$, both far exceeding the natural hadronic scale $\approx 1/m_{\pi }$, and dominant molecular components $X_2>0.95$ under full compositeness. The results are stable for total compositeness $X\in [0.5,1.0]$, where $1-X$ represents an upper bound on any compact/CDD-pole contribution. In the present study, we go beyond the pole-position CDD criterion by explicitly constructing the inverse amplitude as an ERE background plus a scanned Castillejo-Dalitz-Dyson (CDD) pole term, finding no region of physically reasonable bare-state parameter space that reproduces the pole while leaving a natural background; we fully propagate the loop-function subtraction constant $a({\mu }_g)$ as a Monte Carlo nuisance parameter jointly with the experimental mass and width, confirming it contributes a negligible ($<0.01\%$) additional uncertainty; and we test robustness against an assumed correlation between $M_R$ and ${\Gamma }_R$, finding all results stable over a broad range of correlation coefficients. We demonstrate analytically and numerically the scheme-independence of the compositeness coefficients, provide explicit derivations of all ERE formulas from the pole condition, clarify the analytic continuation procedure for closed channels, and compare our results quantitatively with independent ERE, EFT, and other relevant studies. The $Z_c(3900)$ case is treated separately to show the breakdown of the two-channel approximation for above-threshold states. Graphical illustrations of the Monte Carlo distributions are included.