We explore various self-trapped states, including sharp matter-wave solitons and spatially broad self-bound droplets (SBDs), in an effectively one-dimensional atomic-molecular bosonic-condensate mixture (unlike the well-known mechanism of the formation of QDs in atomic Bose-Einstein condensates, in the present system SBDs are produced without the action of the beyond-mean-field effects). The analysis reveals a transition between the solitons and QDs, passing through a self-bound hybrid state (SBHS). When the intermolecular interaction is switched off, the system exhibits a soliton state with low values of the scaled number of particles (alias the total norm), N. The system transitions to the SBHS as N increases. In the hybrid regime, both the maximum density and spatial size of the state grow with N, while the chemical potential decreases. The SBHS family is described, in an analytical form, by the “cascading-limit approximation”, which is suggested by the similarity to the optical second-harmonic-generating system. A phase diagram in the (g,N) plane is constructed, and the Bogoliubov-de Gennes (BdG) analysis together with real-time simulations confirms the stability of all three types of self-bound states in their respective parameter regimes. Simulations demonstrate that collisions between the bound states are inelastic, featuring fusion or fragmentation, depending on the collision speed.