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The A2Mo3As3 (A = K, Rb, Cs) compounds are built of assembled Mo3As3 chains and are thus the potential realisations of quasi-one-dimensional metals. Various experiments on these materials hint to the Tomonaga-Luttinger liquid (TLL) physics from which superconductivity with a relatively high critical temperatures of Tc ≈ 10.5 K emerges. However, TLL and its relevance to emerging superconductivity are not yet resolved for A2Mo3As3 due to their multiband nature.Here we report a combined 75As nuclear quadrupole resonance (NQR), 87Rb and 133Cs nuclear magnetic resonance (NMR) and muon spin relaxation (mSR) study of Rb2Mo3As3 and Cs2Mo3As3 powders. The alkali metal spin-lattice relaxation rates show a characteristic power-law temperature dependence over a broad temperature range, which is a hallmark of the TLL dynamics dominated by the two-quasi-one-dimensional bands. Detailed analysis reveals that these compounds may be in an unusual state of effective attractive interactions, which opens up intriguing possibilities for the unconventional pairing symmetries of the superconducting state. To address the superconducting state we, in addition to 75As and alkali metal relaxation rate studies, employ also the ,transverse-field mSR. The temperature dependence of the muon relaxation rate, s, is in the superconducting state due to the field distribution created by the vortex lattice and is compared against s-, p- , and d-wave scenarios. The penetration depth l = 669 nm and the coherence length z = 3.4 nm are also determined.