The spin-wave response of the $S=\frac{1}{2}$ Ising-like antiferromagnetic chain compounds CsCo${\mathrm{Br}}_{3}$ and CsCo${\mathrm{Cl}}_{3}$ has been studied by inelastic neutron scattering. The asymmetry at low temperatures of the line shape of the spin-wave continuum is not correctly predicted by the existing theory of isolated chains to first order in the transverse exchange interaction. The effective spin Hamiltonian for a chain has been rederived and shown to include a slow internal staggered field from exchange mixing that influences the rapid spin fluctuations. Three-dimensional correlations are shown to be important above but close to N\'eel temperature. The first-order calculation of Ishimura and Shiba has been extended to include these effects and numerical calculations made for $0<T<\frac{J}{{k}_{B}}$. A good account is obtained of the spin-wave continuum and its temperature dependence. The results show that the highly asymmetric line shape arises from local staggered fields which vary on a much slower time scale than the transverse spin motion, and whose effect decreases with increasing temperature in the manner expected for the decay of the intrachain and interchain correlations. An improved and simplified account of the Zeeman ladder as seen in the Raman scattering at low temperatures has also been obtained.