We present a complex study of pulse-energy-controlled solitonic self-switching of femtosecond pulses at wavelengths of 1700 and 1560 nm in two nonlinear high-index contrast dual-core fibers having different levels of slight asymmetry. In the case of the fiber with higher dual-core asymmetry excited by 1700 nm pulses, the highest switching contrast of 20.8 dB at 40 mm fiber length was demonstrated. It was accompanied by multiple exchanges of the dominant core at the fiber output, which is a strong signature of the soliton-based switching process. In the case of the fiber with lower dual-core asymmetry, excited by 1560 nm pulses, the highest switching contrast of 21.4 dB at 35 mm fiber length was achieved with a broadband character of the switching in the spectral range of 1450–1650 nm. Both demonstrations represent progress in all-optical switching studies at these particular wavelengths thanks to a comparison between their results, which reveals the requirement of a higher level of dual-core symmetry for applicable C-band operation.