Are characters not allowed to speak in short sharp (grammatically incorrect) ways? If not why not?

Yes, it could be allowable, but IMO, there are two main problems to be considered:

1. As Kiya, and others, already pointed out, it must be consistent to be believable and form characterization.

2. Even more difficult - and I am saying this as a German native with knowledge and experience with a variety of regional dialects within the German language - those dialects are mainly phonetic, meaning there is no specific spelling nor grammar, when its written. Writing in dialect is terribly hard, as you are using a tool (written language) that is not always adequate to the task.

The issue becomes even more critical, when your readers are not themselves actively familiar with that dialect. What I mean is, if you know the dialect and your "audience" (= readers) do also - then it is easier. If however your intention is to reach people beyond that, then it becomes a matter of possible misunderstanding or plain non-understanding (which actually is like in real life, when listening to colloquial regional dialects. While I fancy my English to be adequate to make myself understood, I did find that not only Americans, but also Scots and Irish are separated from the English by a "common" language).

I have no "set in concrete" view on this, but I recommend to give the balance some consideration. On one side, I (German, who learned English as a foreign language) am indeed irritated at first by the "incorrect phrasing" - on the other side, I do agree, that it adds to the ambiente and the colouring of the character. (I would hate to have to translate something like this, with all its facettes, and for a Germany wide readership!)

In times of crisis it is of the utmost importance not to lose your head (Marie Antoinette)