A Flight Service Station specialist reported they failed to get all the required flight plan information from a VFR aircraft that called them to report a rough running engine.

Narrative

A pilot called over the radio to ask if we had heard anything from Aircraft X as he was having engine issues. A rough engine as he said. During the conversation with the pilot; the pilot of Aircraft X called over another frequency and I let the pilot know and ending my conversation with him. The pilot of Aircraft X told me he had a rough engine and was looking to let someone know as well as file a Flight Plan (FP) in case he was unable to make his destination. At the time of this radio transmission he was on his way to BTT airport. I asked him if he wanted to [request priority] he said no not yet and proceeded to file a VFR flight plan with me. At the time I was nervous as this was my first potential [situation] and I ended up missing some FP information; such as contact information and ETE. I began to try and keep communications with him and get the addition FP information when the Supervisor at the time told me to keep transmissions with this pilot to a minimum so he can focus on the flight. With my adrenalin still high I blindly followed what was said and ultimately forgot to get the remaining information for the FP from him. Around ten minutes later when my nerves were a bit more calm I proceeded to try and start communications with him again however he did not respond. Around another five minutes past when he finally reached out to us letting us know he was safe at BTT and to close out his FP.Suggestion: Additional training on dealing with [priority] situations. While not all situations can be prepared for learning to keep my nerves in check would have helped me keep a clear head and realized that blindly following what supervisor said ultimately ended up causing me to miss key information. Though nothing happened this time; had something happened that information would have been critical in helping locate him.