Does "whether or not I should..." sound awkward?

Solution 1:

It doesn't sound awkward. But the "or not" is superfluous. Obviously, if you don't know whether you should go out with someone, the other possibility is that you don't go out with them. It's perfectly fine either way.

A third possibility, also perfectly fine, is to omit the "or not" but change to the negative, like this:

"He stood me up three times in a row," said Alice angrily. "Now I wonder whether I shouldn't go out with him at all!"

Native speakers would use all three constructions with no particular preferences. Interestingly, you can replace "whether" with "if" in all but the "or not" case. To use "or not" with "if", you have to say "if I should ... or not".

Solution 2:

That's what a native speaker would say. http://www.pearsonlongman.com/ae/azar/grammar_ex/message_board/archive/articles/00095.htm

Solution 3:

Perhaps a better sentence to our ear would be: "He stood me up three times in a row," said Alice angrily. "Now I wonder whether or not to go out with him at all!"