Sawdust from deciduous trees was used as a raw material for the preparation of carbonaceous adsorbents. Microwave-assisted chemical activation with K2CO3 and H3PO4 was used to produce materials with a well-developed porous structure. The obtained activated biocarbons were characterized in terms of their porous structure, elemental composition, morphology, thermal stability, as well as surface and electrokinetic properties. The sorption abilities of both materials towards synthetic (poly(acrylic acid)) and natural (lysozyme) polymers in the process of their removal from aqueous systems were determined. Both single adsorbates and mixed solutions of two polymeric adsorbates were tested. The stability of aqueous suspensions containing activated biocarbons and one or two polymers was also determined. As a result of microwave-assisted chemical activation two carbonaceous adsorbents were obtained, characterized by a very well-developed specific surface area (1093–1777 m2/g), a completely different type of porous structure (mesoporous or microporous), and the acidic nature of the surface. The maximum adsorption of poly(acrylic acid) was obtained from a mixed solution of both polymers and it reached values of 379 mg/g (for the sample activated with H3PO4 with mean pore diameter 3.04 nm and minimal contribution of micropores—0.3%) and 259 mg/g (for K2CO3 activated material characterized by the mean pore diameter equal to 1.72 nm and large contribution of micropores—77.4%). In the case of lysozyme, the adsorption efficiency was two times lower (sorption capacity of 127–166 mg/g). Based on the collective data analysis, it can be stated that the most probable mechanisms of polymeric destabilization (highly desirable in separation from the multicomponent solutions) are surface charge neutralization at pH 3 and bridging flocculation at pH 11 (especially for the systems containing material activated with H3PO4 and poly(acrylic acid)).