Question: Every time there was a murder, Jessica Fletcher wasn't far away. Why was she never suspected in any of the cases?
Answer: There actually was one episode where Jessica was suspected of murdering a Bulgarian diplomat. "J.B. As in Jailbird."
Question: How did frank Die?
Question: Why do some of the episodes have someone else investigating and solving a murder instead of Jessica?
Answer: Because during Season 6, Lansbury wanted a lighter workload and to do other projects. So they had her just introduce and sum up the episode.
Question: Why did later seasons have many of the episodes take place in New York?
Answer: The storyline was that Jessica was teaching classes in New York. However, after so many seasons and as ratings declined, it was probably to freshen up the series with a different locale, introduce new characters, and have a greater variety of situations in which Jessica could solve murders. It was already unbelievable that so many people could be murdered in the small town of Cabot Cove. Being in a large city like New York makes it a tiny bit more plausible.
Chosen answer: This is the very reason my brother and I used to jokingly call the show, "Murder, She Caused." It's amazing she was ever on anyone's guest list for a party, given the likelihood someone would end up deceased. As to your question, most of the time, Jessica Fletcher would have had an air-tight alibi, as she was in a room full of people, or her whereabouts were accounted for when a murder occurred elsewhere. It also seems to me that there were episodes where she, purely with respect to opportunity, could have been a suspect. I believe she even acknowledged that as a logical possibility from time to time, even though she knew, of course, she was not the killer. However, the investigation would obviously rule out the possibility of her involvement, eventually.
Michael Albert