We present a study of ultrafast solitonic all-optical switching in the C-band using dual-core optical fibers fabricated in-house. The two soft glasses used for the fiber express high contrast of refractive index at the level of 0.4. The core was made of a highly nonlinear lead-silicate glass, which, combined with the anomalous dispersion, allowed the solitonic propagation at low pulse energies. A convincing switching performance was demonstrated using the optimized fiber at hundreds of picojoule level of the in-coupled energy. The dedicated numerical simulations unveiled the solitonic propagation character of 75 fs excitation pulses influenced by dual-core asymmetry. All the key parameters of the switching, such as the optimal fiber length (38 mm), the switching contrast (4.5 dB) and the switching energy levels (200–700 pJ), were confirmed by the numerical results.