Altermagnets (AMs) have recently emerged as a distinct magnetic class bridging central features of ferromagnets (FMs) and antiferromagnets (AFMs), offering new opportunities for spin-based electronics. While they possess zero net magnetization like collinear AFMs, they simultaneously exhibit momentum-dependent spin splitting long thought exclusive to FMs. Despite intense theoretical interest, experimentally accessible materials hosting both altermagnetism and nontrivial band topology remain scarce. EuAgAs, crystallizing in space group P6₃/mmc, was previously identified via density functional theory (DFT) as a bulk altermagnetic Dirac semimetal. Contrary to these predictions, our neutron diffraction experiments reveal that the bulk ground state adopts a q = (0,0,1/2) AFM structure with an in-plane ↑↑↓↓ spin sequence. Systematic DFT calculations, however, uncover a remarkable near-degeneracy among competing magnetic orders: the FM and AM configurations lie only 0.11 and 0.40 meV/f.u. above the AFM ground state, respectively. We further show that while a simple Heisenberg model favors a spin-spiral ground state, the inclusion of non-Heisenberg biquadratic coupling stabilizes the observed commensurate AFM phase. This near-degeneracy renders the magnetic state highly tunable, with DFT predicting a transition to the altermagnetic phase under hydrostatic pressure at approximately 14 GPa, establishing EuAgAs as a controllable platform for accessing topological altermagnetism.