Additive manufacturing allows for on-demand production of functional constructs through an accurate layer-by-layer deposition of filaments. These are generally made of polymer-based materials that exhibit viscoelastic features. As the material flows outside the nozzle, die-swell phenomena occur, leading to a loss of printing resolution and a shape fidelity issue in the printed construct. To accurately capture changes in the extrudate shape when the polymer-based material leaves the die, it is essential to address a free-surface problem. Modeling viscoelastic flows with moving boundaries is challenging due to the intricate interplay between elasticity and capillarity. This complexity can result in localized and significant changes in surface curvature, as well as sharp variations in stress distribution. Addressing these complexities requires robust numerical modeling techniques capable of capturing the evolving free-surface behavior during extrusion. Furthermore, in coaxial extrusion, an emerging technique for producing filament architectures with core-shell structures, the modeling challenge extends to a multiphase framework. The interaction between the inner and outer fluid layers introduces additional interfacial dynamics, requiring careful treatment of immiscible phase boundaries and rheological contrasts. Accurate numerical representation in such settings necessitates advanced computational strategies capable of resolving multiphase flow and interfacial stresses. In this study, the extrusion stage in additive manufacturing is analyzed through a finite element methodology grounded in the Arbitrary Lagrangian-Eulerian (ALE) framework. The complex free-surface and fluid-fluid interface dynamics are addressed by numerically resolving the isothermal, incompressible form of the Cauchy momentum equations, in conjunction with differential constitutive models that describe viscoelastic behavior. The ALE approach facilitates accurate resolution of moving interfaces, thereby capturing the evolution of material flow and surface deformation as the filaments exit the nozzle. The computational outcomes derived from this strategy offer critical insights into how rheological material characteristics and processing conditions govern the extrudate shape.