UTe₂ is a recently discovered unconventional superconductor that has attracted much interest because of its potentially spin-triplet topological superconductivity. Our ac calorimetry, electrical resistivity, and x-ray absorption study of UTe₂ under applied pressure reveals key insights on the superconducting and magnetic states surrounding pressure-induced quantum criticality at P_c1 = 1.3 GPa. First, our specific heat data at low pressures, combined with a phenomenological model, show that pressure alters the balance between two closely competing superconducting orders. Second, near 1.5 GPa, we detect two bulk transitions that trigger changes in the resistivity, which are consistent with antiferromagnetic order, rather than ferromagnetism. Third, the emergence of magnetism is accompanied by an increase in valence toward a U⁴⁺ (5f²) state, which indicates that UTe₂ exhibits intermediate valence at ambient pressure. Our results suggest that antiferromagnetic fluctuations may play a more substantial role on the superconducting state of UTe₂ than previously thought.