Therefore is an adverb, as well as hence and thus, although used in this similar manner they are conjunctive because they denote causal relation between the two clauses in your statement.
Both of your examples above are acceptable, but there should be a semicolon in your first therefore example:
I was tired; therefore, I fell asleep.
No, therefore should not be reserved for conclusions beyond a reasonable doubt. It is merely a transition similar to thus or as such:
Therefore, I ordered pepperoni.
Thus, I ordered pepperoni.
As such, I ordered pepperoni.
The extended conversation could have been:
I like meat on my pizza. Therefore, I ordered pepperoni.
You can test for an appropriate use of therefore by flipping the sentence order and using because:
I ordered pepperoni because I like meat on my pizza.
The contention that the therefore segment should be restricted to perfect logical use is forgetting that the word really only serves to draw a causal link between statements.
I like red. Therefore, I painted my wall red.
There is no logic here. I am just explaining why I painted the wall red.
Now, if someone is using therefore as a logical link and the logic sucks, you can say that the argument is bad. But the use of therefore isn't the problem.
This $10 item is 50% off; therefore I am saving $6.
This is wrong, but the use of therefore isn't incorrect because it is simply communicating the thought. The communication is accurate; that makes therefore the appropriate word. Replacing therefore with a different word changes the meaning of the sentence (and could correct the logic) but the intent of the speaker no longer matches the communication.
Edit: Since there seems to be some confusion about the actual definition of the word, here is what my dictionary says:
for that reason; consequently : he was injured and therefore unable to play.
Reason, in this context does not mean "logic". It just means "why".
Why couldn't he play? / He was injured.
He was injured and therefore unable to play.
He was unable to play because he was injured.
It is worth noting that there is a strict logical use for the term therefore that explicitly means something akin to "logically derived from the previous statements" but that would be applicable to formations of the following:
All men are mortal
Aristotle is a man
Therefore, Aristotle is mortal
This is commonly represented by three dots in a triangle (∴). But even in this case, the use of therefore is a signal of a specific meaning. If the conclusion is false, it was not an incorrect use of therefore but simply faulty logic. Removing or changing the word doesn't make the problem go away.
Best Answer
In addition to the methods presented in RegDwigнt's answer — using synonyms and simply removing the words — two other techniques for avoiding such repetition come to mind.
Replacing "thus" or "therefore" in the conclusion with "since" or "because" in the clause presenting the evidence is perhaps the most straightforward. This technique works better with shorter statements of evidence.
The second technique would be to use a verb that includes the logical inference such as "indicate", "imply", "show", "demonstrate", or even "allows". Using a participle avoids the need for an explicit subject, but "this" or "such evidence" are adequate subjects. (In fact, "such evidence" might be preferred when the evidence is extensive.)
This can include reversing the order of evidence and conclusion.
(The latter rephrasing is clearly problematic since the colon indicates that a stronger binding to "statistical inference problem" is desired. However, it provides the useful example of "recognize" replacing "show", changing the subject from the evidence to the observer allows further variety in verbs.)