We investigate eigen oscillations of internal degrees of freedom (Higgs mode and Goldstone mode) of two-band superconductors using generalization of the extended time-dependent Ginzburg-Landau theory, formulated in a work Grigorishin (2021) [1], for the case of two coupled order parameters by both the internal proximity effect and the drag effect. It is demonstrated, that Goldstone mode splits into two branches: common mode oscillations with acoustic spectrum, which is absorbed by gauge field, and anti-phase oscillations with energy gap (mass) in spectrum determined by interband coupling, which can be associated with Leggett mode. Analogously, Higgs oscillations splits into two branches also: massive one, whose energy gap vanishes at critical temperature Tc, another massive one, whose energy gap does not vanish at Tc. It is demonstrated, that the second branch of Higgs mode is nonphysical, and it with Leggett mode together can be removed by special choice of coefficient at the "drag" term in Lagrangian. In the same time, such choice leaves only one coherence length, thereby prohibiting so-called type-1.5 superconductors. We analyze experimental data about Josephson effect between two-band superconductors. In particular, it is demonstrated, that the resonant enhancement of the DC current through a Josephson junction at a resonant bias voltage Vres, when the Josephson frequency or its harmonics match the frequency of some internal oscillation mode or its harmonics in two-band superconductors (banks), can be explained with the coupling between AC Josephson current and Higgs oscillations. Thus, explanation of the effect does not need Leggett mode.