This is an excellent question, which brings into focus many challenging questions of modern philosophy. As a spiritual dualist myself, I have trod this path, and can offer pointers.
First, and an aside, the reference to "extraordinary claim", is a red herring, and represents a fallacy which must be disposed of. To do empiricism properly, there is no "ordinary" vs. "extraordinary" distinction in observations. Nor in inferences. Anyone objecting that either an observation or inference is "extraordinary" and demand overwhelming support before being considered, is engaged in the fallacy of wishful thinking, then specials pleading to raise the bar for an observation or inference they so not want to accept. Probe what they mean by "extraordinary" and most who use the term will ultimately admit they mean "non-material". And as we all experience non-material qualia (see "The Hard Problem of Consciousness" for how qualia are non material), and non-material abstract objects and information every day, calling non-material things "extraordinary" is a gross misuse of the term for rhetorical purposes.
What is relevant in empiricism, and a valid criteria which the "extraordinary" language misrepresents, is that observations and inferences which contradict well-established certainties, DO have higher bars to acceptance than those which do not. "Facts" are just observations or well-established inferences we have high confidence in. But that "the world is ontologically solely material" is NOT a well-established certainty, and neither observations nor inferences that contradict it are subject to any "higher bar" standards.
What we DO know, is that the material world mostly seems to operate without non-material interdiction, with notable exceptions of a) consciousness on living things, b) math on fundamental physics, c) ideas on human thought and society, and d) relational abstractions (relationships, causation, time sequencing) affecting ALL material interaction. These are a large and diverse enough suite of exceptions, that the bar to expand them, while somewhat higher than for phenomena that don't contradict them, is not justifiably particularly high.
Now to your example: "I went to sleep, but then I suddenly woke up at 3:00 a.m. feeling a terrifying evil presence, and then my bed began to shake violently" This example has a sequence:
- woke 3 am
- felt terrifying presence
- bed shook
None of these in themselves contradict what we know, nor does the sequence in combination, so no, this is not a good example of something "extraordinary". The INFERENCE to "there was an evil entity that I could sense which shook the bed" -- THAT has a higher bar to be accepted. However, note that the ability to sense morality -- is not contradicting what we know, and the ability of conscious entities to causally affect our world is likewise not contradicting what we know. The only noteworthy new assertion is that such an entity could be discarnate.
I still have not gotten to your core question, but I have one more tangent to address first, and that is first person empiricism. Humans are naturally empiricists. when trying to crawl, or walk, children try variants on motion, then gradually revise how they move based on observed successes or failures. A more refined process was that of learning how to make pots, or fletch arrows. This is primarily a first person process -- we humans do almost all of our empiricism in the first person. The problem with empiricism is not that one uses the first person, but that we humans are particularly poor at resisting confirmation bias. When one is just doing empiricism by oneself, there is no peer community to point out our blind spots in compiling evidence, or analyzing it, or considering alternative explanations. To do first person empiricism well, one must actively TRY to refute the conclusion one prefers, AND look to observations collected in more challenging circumstances than the ones that support one's conclusion. 3rd person empiricism has an advantage over first person, in that one is at least confirming the basic observation, and therefore excluding most instances of delusion.
Note that all observations are first person. 3rd person empiricism just is looking at confirmation from other first person observers. This final point lets me answer your question -- YES, having multiple observers report the same phenomenon DOES increase the credibility of the observation.
I will still challenge one more part of your question -- the inference to "more likely to be true" is a mis-statement of what empiricism provides. It references an absolutist "truth" rather than the pragmatic "utility". We cannot quantify "likelihood of truth", as we have no access to "truth". What we have is USEFULLNESS. And an observation or inference that applies to many people, and across multiple circumstances, would be more usefully generalizable than one that is based on a single instance or particular type of circumstance.