Radio observations of a few cool-core galaxy clusters have revealed the presence of diffuse emission on cluster scales, similar to what was found in merging clusters in the form of radio halos. These sources might suggest that a minor merger, while not sufficiently energetic to disrupt the cool core, could still trigger particle acceleration in the intracluster medium on scales of hundreds of kiloparsecs. We aim to verify the occurrence of cluster-scale diffuse radio emission in cool-core clusters and test the minor merger scenario. With the LOw Frequency ARray (LOFAR) at 144 MHz, we observed a sample of twelve cool-core galaxy clusters presenting some level of dynamical disturbances, according to X-ray data. We also performed a systematic search of cold fronts in these clusters, re-analysing archival Chandra observations. The clusters PSZ1G139.61+24, A1068 (new detection), MS 1455.0+2232, and RX J1720.1+2638 present diffuse radio emission on a cluster scale ($r $). This emission is characterised by a double component: a central mini-halo confined by cold fronts and diffuse emission on larger scales, whose radio power at 144 MHz is comparable to that of radio halos detected in merging systems with the same cluster mass. The cold fronts in A1068 are a new detection. We also found a candidate plasma depletion layer in this cluster. No sloshing features are found in the other eight clusters. Two of them present a mini-halo, with diffuse radio emission confined to the cluster core. We also found a new candidate mini-halo. Whereas, for the remaining five clusters, we did not detect halo-like emission. For clusters without cluster-scale halos, we derived upper limits to the radio halo power. We found that cluster-scale diffuse radio emission is not present in all cool-core clusters when observed at a low frequency, but it is correlated to the presence of cold fronts. The coexistence of cluster-scale diffuse radio emission and cold fronts in cool-core clusters requires a specific configuration of the merger and so it puts some constraints on the turbulence, which deserves to be investigated in the future with theoretical works.