PA-28 pilot reported a vibrating engine oil access door; decreased cylinder head temp on one cylinder and a change to engine sound during climb which were caused by a sheared exhaust manifold.

Narrative

I was flying cross country from ZZZ to ZZZ1. I advanced the power to initiate a climb from 5500 to 7500 MSL. There was an audible change in the engine noise with a distinct rattle plus my EGT sensor on the #3 cylinder fell offline. My initial reaction was to monitor my other engine instruments and to check for my nearest available airfield. The engine was performing fine; except for the sensor issue; and I saw what was causing the noise. The oil access door on the cowling was rattling. With normal engine indications; I elected to continue to my home airfield instead of finding a near by field to divert to and investigate the issue. After landing; I pulled the cowling off and discovered that the exhaust tube that connects the cylinder to the muffler had sheered completely. What I thought was just a loose oil access door; was actually an indication that the exhaust from cylinder #3 was venting directly into the engine compartment; which was blowing hard enough to move the oil access door despite the wind flow holding it down. My lack of experience with this type of failure and insufficient systems knowledge of the exhaust system decreased my situational awareness of what was happening with the airplane. I coupled that with a want to get home and not be stuck at a divert location. On reflection; this combination convinced me that there wasn't an issue with the engine and I could continue flying the airplane. This was not the right call; I was lucky that the failure didn't cause a bigger issue like an engine fire or failed in some other way. In the future; I now have the SA to understand that a rattling oil access door is an indication of a significant issue going on inside the cowling. I can then make a better decision about diverting and investigating the problem.