You're asking whether well can be used attributively (as in "a well person") and not just predicatively ("she is well"). A great question!
The Cambridge Grammar of the English Language (CGEL) on page 560 includes well among those adjectives that do not appear attributively, with only a few exceptions. Here's what they wrote:
Well is used attributively in the construction He's not a well man, but in general it is excluded from attributive use: compare *his well mother.
The exception mentioned in CGEL is specific to the negative phrase is not a well X, where X refers to a type of animate being (typically a person, though an animal is also possible). Here are some examples from The Corpus of Contemporary American English (COCA):
- Just an opinion, but he's not a well person.
- Bear in mind how I'm not a well woman.
- A plan was also floated to run him as Taft's vice presidential candidate, on the presumption Taft was not a well man and quite likely to die in office.
- I'm not a well woman, Mary.
- Mr. Cheskis, that orange cat is not a well cat. It should go to the ASPCA.
And you will find a few similar exceptions, including phrases like "a well-woman visit" where well is apparently an attributive modifier in a larger attributive phrase, but in general well does not appear attributively. What's more, these few exceptions we do find aren't particularly common. If you'd like to remember the simplified rule never use well attributively, it would serve you just fine.
Your phrase ?a well-being person is unnatural. Well-being, generally speaking, will be taken as a single noun, and it won't be understood the way you intended.
In this answer, the * symbol indicates that a phrase is ungrammatical, while the ? symbol indicates that a phrase is questionable.