Direct air capture and CO₂ conversion will play vital roles in a future circular carbon economy. Here, we propose a novel system for integrated CO₂ capture and conversion through the combined use of solid and liquid sorbents and demonstrate its technical feasibility. CO₂ is initially captured from the air using an amine-functionalized solid sorbent that is regenerated by rinsing with an aqueous carbonate-rich solution. The resulting bicarbonate-rich solution is fed to an electrolyzer, converting the bicarbonate electrolyte to syngas. Through our experiments, we demonstrate that the dual-sorbent system is capable of capture from air as well as solid sorbent regeneration and CO₂ conversion. We use a simplified process model based on experimental results to explore the effects of scaling the components of the proposed system. We find that the pH-swing between the rich and lean solutions is a dominant design parameter, which is almost exclusively governed by the electrolyzer sizing. A small pH swing results in an improved electrolyzer performance and beneficial syngas composition, whereas a large pH swing results in efficient solid sorbent use and decreased water loss. Our results further highlight the fundamental trade-offs that are present when designing integrated capture and conversion systems.