We present an experimental study of double-stranded DNA diffusion in slitlike channels. The channel heights span the regime from moderate confinement (height similar to bulk radius of gyration of the DNA) to strong confinement (height similar to persistence length). Scalings of diffusivity with channel height differ from blob model predictions. The diffusivity scales inversely with molecular weight when the channel height is smaller than the bulk radius of gyration. This scaling is indicative of hydrodynamic screening. A scaling analysis shows that the screening of hydrodynamic interactions arises from a combination of two mechanisms. After using a Zimm preaverage approximation, the unique symmetry of the thin-slit disturbance velocity and the isotropic nature of the polymer conformation together cause a cancellation of hydrodynamic interactions due to symmetry. We also find that the algebraic decay of the far-field velocity magnitude is sufficient to eliminate large-length scale hydrodynamic cooperativity in diffusion of quasi-two-dimensional polymers in good solvents.