We present radio and X-ray studies of A3444 and MS1455.0+2232, two galaxy clusters with radio minihalos in their cool cores. A3444 is imaged using the Giant Metrewave Radio Telescope (GMRT) at 333, 607, and 1300 MHz and the Very Large Array at 1435 MHz. Most of the minihalo is contained within r < 120 kpc, but a fainter extension, stretching out to 380 kpc southwest of the center, is detected at 607 MHz. Using Chandra, we detect four X-ray sloshing cold fronts: three in the cool core at r = 60, 120, and 230 kpc, and a fourth one at r = 400 kpc—in the region of the southwestern radio extension—suggesting that the intracluster medium (ICM) is sloshing on a cluster-wide scale. The radio emission is contained within the envelope defined by these fronts. We also analyzed archival 383 MHz GMRT and Chandra observations of MS 1455.0+2232, which exhibits a known minihalo with its bright part delineated by cold fronts inside the cool core, but with a faint extension beyond the core. Similarly to A3444, we find a cold front at r ∼ 425 kpc, containing the radio emission. Thus the entire diffuse radio emission seen in these clusters appears to be related to large-scale sloshing of the ICM. The radio spectrum of the A3444 minihalo is a power law with a steep index α = 1.0 ± 0.1. The spectrum steepens with increasing distance from the center, as expected if the minihalo originates from reacceleration of relativistic particles by the sloshing-induced turbulence in the ICM.