The correct option is A either, or
The given sentence expresses a choice between two possibilities "attempt to solve the problem" and "resign from her post". Only one choice can be made, as they're both alternatives, hence the pair of conjunctions 'either...or' is appropriate, because it expresses a choice between two alternatives. Hence option A is correct. 'Though, yet' refers to a contrast between the possibilities, where they're opposite to one another and no choice is made, they're both true in spite of being opposite in meaning, whereas here there isn't any contrast, but two alternative choices. Hence 'though...yet' isn't appropriate and option B is incorrect. 'Whether, or' refers to a choice between two possibilities, just like 'either, or', but there's a slight difference. 'Whether, or' is used in indirect speech, it also means that 'it doesn't matter' which choice is chosen, whereas the given sentence is in direct speech and presses on the importance of the options. Hence, option C is incorrect. 'Both, and' is used when both the possibilities are chosen, whereas here there is a choice, one cannot solve the problem as well as resign, because resign is avoiding the problem altogether, hence option D is incorrect.