PA-28 pilot reported smoke in the cockpit during flight and diverted to a safe landing. Pilot stated they experienced symptoms of carbon monoxide poisoning.

Narrative

During a routine part 91 flight; I experienced smoke in the cockpit. I immediately ran my checklist and began a climb with a turn towards ZZZ and advised ATC. The smoke dissipated when I turned the landing light off however a strong burning smell was still being emitted. I asked ZZZ tower if they spotted any smoke coming from the aircraft to which they replied they did not. I asked for Runway XXR instead the original XY I requested and was cleared to land. Upon landing I was met by ZZZ ARFF (Aircraft Rescue and Fire Fighting) and they verified no fluid was leaking and no smoke was visible. I taxied to the FBO and conducted a post flight inspection. The exhaust manifold appeared to be charred and damaged as well as the orange vent on the copilot side of it. Burn marks appeared on the lower exterior cowling vent was noticeable. After the situation; I felt uneasy with a moderate headache; dizziness; and symptoms of carbon monoxide poisoning. I was treated at a hospital for Carbon Monoxide poisoning.