The first and most important thing to do is research the Gettysburg Address itself. There are several pages of history right here on eNotes, as well as entire books and websites dedicated to the historic event; some quick research on Google should give you all the information you need for historical accuracy.
Second, try to think of a suspenseful or scary theme to the story that does not contradict the existing history -- unless you are writing an "alternate history" story about the event, in which case feel free to change the events as necessary. Good topics would include the end of the Civil War, the Lincoln Assassination (two years later), the very recent events of abolition, and the Address itself, which became a national event of historic proportions. For example, you could write a story about the five existing original copies of the address, but a sixth copy was stolen to use as Confederate propaganda. Another good plot could concern the people who actually heard the Address; they might fall victim to a curse later in life, or perhaps rise up and attack the podium. If you need a fast plot idea, you could take some generic ideas and incorporate them into the history of the Address.
For dialogue, don't try to emulate grammar or syntax of the era; unless crafted by a Mark Twain, phonetic grammar tends to sound patronizing. Instead, just write out clear sentences in plain English, styled to the education level of the speaker; a former slave or working farmer would use fewer large words or complex grammar than a senator or plantation owner. Don't use "saidisms," or words in place of "said," such as "exclaimed" or "shrilled" except in rare cases when such a word would add extra effect; similarly, don't use adverbs like "nastily" to modify "said" except when it helps the flow of the sentence. In fact, in cases when there are only two or three people talking, you can leave out any identifiers altogether; it should be clear from the dialogue itself who is talking.
Finally, make sure you edit and revise a few times to remove clunky phrasing or missed errors in time, place, and continuity. Have someone else read over the story and offer constructive criticism. Try editing the story backwards, sentence-by-sentence. Above all, try to tell a story that you would enjoy reading.