Actually, "being sick" has the implication of 'throwing up', as in "During the entire boat trip, John was being sick," or "I was being sick all over the table". "I have been sick" means that you have been unwell, or ill, due to a disease.
They're both perfectly alright, they just mean differently.
Edit: In lieu of Rolfer's change of question:
Let's take the adjective "happy". The two examples are now "I have been happy" and "I have been being happy".
The first of the two sentences means that you were happy in times past. You might not necessarily be happy now, but you were happy before.
The second of the sentences however, meant that you have been doing some activity which occurs when you are happy, for example, Dancing, or jumping, or shouting, or singing, etc. "I have been being happy" denotes that you were doing an action associated with happiness. This can also be seen by "sick", or "angry" ('being angry' could mean stamping your foot, shouting, saying bad things, etc.).
Essentially, they mean the same thing, just stating it in a different way.
The first two are grammatical. In the first, married is the past tense of marry, and is here used intransitively.
In the second, they were married is a passive construction, and married is the past participle of marry. Although it is passive, the agent is not mentioned, but we can reasonably assume that they were married by a priest, or by a government official.
The third is ungrammatical because the past perfect construction describes a past event that precedes another. It is clear in this case that the wedding took place only after they had been engaged, and not before.
Best Answer
The first sentence is fine. It can be given some journalistic oompah though by using a participial phrase. Like this:
Not like this:
For same timespans, use present participle (the one ending in -ing). To place something in an anterior timespan, use past participle. A few more examples of the former: