We present a comprehensive muon spin relaxation/rotation (μSR) study of the Co²⁺-based alternating Heisenberg 𝐽_eff=12 spin chain system Sr2⁢Co⁢(SeO₃)₃. Low-temperature magnetic property measurements confirm a spin-singlet ground state and identify two critical fields, 𝐻C1 and 𝐻_C2. At 𝐻_C1, the system evolves from the spin-singlet state to a canted antiferromagnetic phase, which persists up to 𝐻_C2, where the system enters a fully polarized state. This magnetically ordered phase exists between μ0⁢𝐻^⊥⁢𝑐_C1=2 T and μ0⁢𝐻^⊥⁢𝑐_C2=3.05 T for μ₀⁢𝐻⁡⊥⁢𝑐 and between μ₀⁢𝐻^//⁢𝑐_C1 = 2 T and μ0⁢𝐻^//⁢𝑐_C2 = 3.6 T for 𝐻⁡ // ⁢𝑐, reflecting the presence of anisotropy. Additionally, μSR results provide insights into the thermal and field evolution of spin dynamics, revealing a thermal crossover in spin correlations, the reopening of a spin gap due to anisotropy above 𝐻_C1, and the emergence of a magnon gap above 𝐻_C2. These findings imply the suppressed criticality of the canted antiferromagnetic phase by anisotropy. Our study establishes Sr₂⁢Co⁢(SeO₃)₃ as a model system for investigating field-driven quantum phase transitions in one-dimensional 𝐽_eff=1/2 quantum spin chains with anisotropy.