The latest research on laser beam fusion cutting (LBFC) with static beam shaping have shown a limitation in the quality of cut parts for thick steel plates (> 6 mm) when using solid state lasers. The approach of dynamic beam oscillation has recently shown to be capable of overcoming this challenge, allowing to increase the cutting speed as well as improving cut edge quality beyond the state of the art. The present paper investigates the influence of longitudinal, linear beam oscillation in LBFC of 12 mm mild and stainless steel plates by analyzing different parameters as cutting speed, burr, surface roughness, heat affected zone (HAZ), and recast layer. Reasons for the observed process improvements compared to static beam shaping have been discussed. The adjustment of the energy deposition and interaction time of the laser beam with the material found to be most relevant for optimizing the LBFC process. In particular, for beam oscillation, a gradual energy deposition takes place and increases the interaction time. This reduces the heat input in terms of HAZ and recast layer by more than 50%, resulting in high cut edge quality and more than 70% faster cutting speed.