Numerous studies have confirmed the coexistence of oxidative stress and inflammatory processes. Long-term inflammation and oxidative stress may significantly affect the initiation of the neoplastic transformation process. Here, we describe the synthesis of a new series of Mannich base-type hybrid compounds containing an arylpiperazine residue, 1,3,4-oxadiazole ring, and pyridothiazine-1,1-dioxide core. The synthesis was carried out with the hope that the hybridization of different pharmacophoric molecules would result in a synergistic effect on their anti-inflammatory activity, especially the ability to inhibit cyclooxygenase. The obtained compounds were investigated in terms of their potencies to inhibit cyclooxygenase COX-1 and COX-2 enzymes with the use of the colorimetric inhibitor screening assay. Their antioxidant and cytotoxic effect on normal human dermal fibroblasts (NHDF) was also studied. Strong COX-2 inhibitory activity was observed after the use of TG6 and, especially, TG4. The TG11 compound, as well as reference meloxicam, turned out to be a preferential COX-2 inhibitor. TG12 was, in turn, a non-selective COX inhibitor. A molecular docking study was performed to understand the binding interaction of compounds at the active site of cyclooxygenases.