Dynamic magnetic properties in perpendicularly exchange-coupled [Co/Ni]/Cu (t = 0-2 nm)/TbCo structures show strong dependences on the interfacial antiferromagnetic strength J, which is controlled by the Cu interlayer thickness. The precession frequency f and effective damping constant α of a [Co/Ni] multilayer differ distinctly for parallel (P) and antiparallel (AP) magnetization orientation states. For samples with a thin t, f of the AP state is apparently higher, whereas α is lower than that in the P state, owing to the unidirectional exchange bias effect (H) from the TbCo layer. The differences in f and α between the two states gradually decrease with increasing t. By using a uniform precession model including an additional H term, the field-dependent frequency curves can be well-fitted, and the fitted H value is in good agreement with the experimental data. Moreover, the saturation damping constant α displays a nearly linear correlation with J. It decreases significantly with J and eventually approaches a constant value of 0.027 at t = 2 nm where J vanishes. These results provide a better understanding and effective control of magnetization dynamics in exchange-coupled composite structures for spintronic applications.