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We use imaging from the first three years of the Dark Energy Survey to characterize the dynamical state of 288 galaxy clusters at $0.1 \lesssim z \lesssim 0.9$ detected in the South Pole Telescope (SPT) Sunyaev-Zeldovich (SZ) effect survey (SPT-SZ). We examine spatial offsets between the position of the brightest cluster galaxy (BCG) and the center of the gas distribution as traced by the SPT-SZ centroid and by the X-ray centroid/peak position from Chandra and XMM data. We show that the radial distribution of offsets provides no evidence that SPT SZ-selected cluster samples include a higher fraction of mergers than X-ray-selected cluster samples. We use the offsets to classify the dynamical state of the clusters, selecting the 43 most disturbed clusters, with half of those at $z \gtrsim 0.5$, a region seldom explored previously. We find that Schechter function fits to the galaxy population in disturbed clusters and relaxed clusters differ at $z>0.55$ but not at lower redshifts. Disturbed clusters at $z>0.55$ have steeper faint-end slopes and brighter characteristic magnitudes. Within the same redshift range, we find that the BCGs in relaxed clusters tend to be brighter than the BCGs in disturbed samples, while in agreement in the lower redshift bin. Possible explanations includes a higher merger rate, and a more efficient dynamical friction at high redshift. The red-sequence population is less affected by the cluster dynamical state than the general galaxy population.