Polyurethane (PU) is widely used in medical products due to its biocompatibility and mechanical properties. Electrospinning (ES) was employed to produce PU-based scaffolds intended for cardiovascular devices (CVD) from blends of Carbothane (Carb) with human serum albumin (HSA), dimethylacetamide (DMA), and drugs. Sirolimus (SRL)—an immunosuppressive/anti-proliferative drug—and diclofenac (DF)—a nonsteroidal anti-inflammatory drug—were introduced into ES blends to produce drug-enriched scaffolds that prevent inflammation and cell overgrowth. The biocompatibility, stability, and mechanical properties of the scaffolds and SRL release were studied. The scaffolds possessed good mechanical properties and were stable in PBS and blood plasma (BP) for 120 days. The minimal SRL release rate was observed for the scaffold 3%Carb/10%HSA/DMA/SRL. A study of scaffold interaction with blood demonstrated good hemocompatibility of most scaffolds. A study of human gingival fibroblasts, endothelial cells (HUVEC and EA.hy926), and vascular smooth muscle cell interaction with scaffolds in vitro demonstrated variability in cell viability and pro-inflammatory interleukin IL-6 secretion, depending on both the scaffold composition and the cell type. The incorporation of DF into scaffolds decreased the concentration of IL-6 in the culture medium. The scaffold 3%Carb/10%HSA/DMA/SRL is the best choice for CVD in terms of hemocompatibility, endothelialization, and the induction of minimal inflammation.