We studied the magnetic properties, in particular dynamics, of the correlated spins associated with natural defects in the organic spin chain compounds ($o$-DMTTF)$_2X$ ($X =$ Br, Cl) by means of electron spin resonance (ESR) spectroscopy. Both materials exhibit spin-Peierls transitions at temperatures around 50 K [Foury-Leylekian et al., Phys. Rev. B 84, 195134 (2011)], which allow a separation of the properties of defects inside the chains from the magnetic response of the spin chains. Indeed, continuous wave ESR measurements performed over a wide temperature range evidence the evolution of the spin dynamics from being governed by the spins in the chains at elevated temperatures to a low-temperature regime which is dominated by defects within the spin-dimerized chains. In addition, contributions of triplon excitations to the ESR response below the transition temperature were observed which provides a spectroscopic estimate for the spin-gap of the studied systems. Moreover, details of spin dynamics deep in the spin-Peierls phase were investigated by pulse ESR experiments which revealed Rabi-oscillations as signatures of coherent spin dynamics. We discuss the results obtained from these complementary methods of ESR spectroscopy in terms of solitons localized at the defect sites within the chains. From a comparison of the characteristic damping times of the Rabi oscillations with measurements of the spin relaxation times by means of primary-echo decay and CPMG methods it becomes evident that inhomogeneities in local magnetic fields strongly contribute to the soliton decoherence.