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.
... you can't present perfect (or continuous) and past simple within a sentence.
As it stands, this rule is incorrect. In many cases it is acceptable and logical to mix past and present references in consecutive clauses
I lost my keys last week, but now I have found them.
This makes sense: A was true then, but B is true now.
This, however, does not make sense:
He has decided to go hiking, so I went hiking as well.
This sentence amounts to A was true then, because B is true now. The simple past describes a past event, your going hiking, but what the present perfect describes is not a past event, his decision, but a present state which is the result of a past event--his state of having decided. That present state cannot be the cause of the past event. The cause must be either a past event or a past state:
He decided (event) to go hiking, so I went hiking as well or
He had decided (state) to go hiking, so I went hiking as well.
The important thing is not to mix time references illogically.
As for the sentence in your friend's email:
Getting that email was such a pleasant surprise, because I was just thinking how I've been wanting to send you an email
There is no mixture of time references here, because the progressive construction "I have been wanting" marks a state, not an event, which may very reasonably be taken to continue into the present out of a past which is marked (by "just") as immediate. In effect, these pasts inhabit the same time frame as the present.
In any case, the "rules" are very loosely applied in informal discourse; see my discussion here. A casual email, which your friend probably dashed off in excitement, should not be held to the formal literary standards of coherence.
Best Answer
I was working is certainly a grammatical answer to the question, but whether you say that or I worked depends on the relationship between the speakers, what has gone on previously in the conversation and on what impression you’re trying to give.
If you are asked What were you doing yesterday? or What did you do yesterday? you won’t often reply I was working or I worked. What were you doing yesterday?, for example, can imply a suspicion, or even an accusation. In that case, the reply might well be What do you mean? I wasn’t doing anything yesterday. Similarly, if you’re asked What did you do yesterday? you might reply Um, let me see. First I went to buy a newspaper, then I took the dog for a walk . . .
There can be no single answer to a question like this. Context determines everything.