"Star" morphologies of charged liquid droplets are distinct droplet conformations that, for a certain charge squared to volume ratio, have lower energy than their spherically shaped analogues. For these shapes to appear, the charge should be carried by a single ionic species. A typical example of a charge carrier that we employ in this study is a fully charged double-stranded oligodeoxynucleotide (dsDNA) in an aqueous and an acetonitrile droplet. We-characterize the structure and dynamics of the star-shaped droplets. We find that by increasing the charge squared to volume ratio, the droplet evolves from spherical to "spiky" shapes, by first passing from droplet sizes that undergo enhanced shape fluctuations relative to those of the larger spherical droplets. These fluctuations mark the onset of the instability. We also find that in-the spiky droplet, the orientation of the solvent molecules in the first shell about the dsDNA is very close to that in the bulk solution. However, this orientation is substantially different farther away from the dsDNA. With regards to dynamics, the motion of the spikes is reflected in the autocorrelation functions of rotationally invariant order, parameters that show a damped oscillator form of decay, indicative of the elastic motion of the spikes. We compare the formation of spikes with that of the ferrofluids and the dielectric materials in an electric field, and we conclude that they represent a different entity that deserves its own characterization. The study provides insight into the manner in which the charge distribution may give rise to well-controlled droplet morphologies and calls for experiments in this direction.