We report magnetic studies of a family of triangular quantum Heisenberg antiferromagnets (TQHAFs) with weak additional Dzyaloshinskii-Moriya interaction, Cu-2(OH)(3)(CmH2m+1COO), m = 7, 9 and 11. The three compounds have nearly identical magnetic properties, despite the large differences in the interlayer separation, indicative of 2D behavior. While the high temperature dc susceptibility data suggest TQHAF behavior, at low temperatures the deviations from the TQHAF predictions imply a canted antiferromagnetic type of ordering. consistent with both the strong peak in the second harmonic of the nonlinear ac susceptibility (at approximate to 20 K) and the low saturation magnetization observed in the low temperature hysteresis curves. We propose that the interplay of Heisenberg and Dzyaloshinskii-Moriya exchanges leads to an unusual state with both 2D Ising-like canted antiferromagnetic and spin glass-like characteristics, a state in which order and disorder appear to coexist.