This paper generalizes the treatment by Dubinin et al. <2003> on solitons, oscillitons, and stationary waves in a two ion species (protons and alpha particles) cold magnetized plasma to include the effects of plasma thermal pressures. A fully nonlinear approach is used. For waves propagating transverse to the magnetic field the system of equations again reduces to a first-order differential equation which may be analyzed by the method of phase portraits and the necessary conditions for solitons and periodic waves found. Solutions exist in specific windows of the sonic and Alfvenic Mach numbers. The inclusion of pressure effects leads to the appearance of singularities or critical Mach numbers which lie on certain loci of the speeds of the species. For waves propagating obliquely to the magnetic field, another class of solutions appears, which we call oscillitons, since they exhibit solitary as well as oscillatory wave properties. Existence windows in parameter space (speed and propagation angle and ion and electron betas) in which solitons and oscillitons may be constructed are found from the dispersion equation for stationary waves. Possible applications of the results to different space plasma phenomena are discussed.