(A) Mrs. Applegate said that she doubted whether it would be available.
Sentence A is correct because Mrs. Applegate's words are being paraphrased. Using the word "that" to connect the dialogue tag "Mrs. Applegate said" with "she doubted whether it would be available" indicates to us that what she said is a paraphrase. Also, we can assume Mrs. Applegate would not be talking about herself in the third person by using the pronoun "she". This is a further indication that her exact words were NOT "that she doubted it would be available."
If Mrs. Applegate had actually said "that she doubted it would be available." The correctly punctuated sentence is D. There is a comma after the dialogue tag. The first word of the dialogue should be capitalized, and the entire dialogue must be in quotations, including the period at the end of the sentence.
