Why does a RNN network output different based on the training sequence
RNNs are pattern recognition tools. It is't entirely clear to me what it is exactly that you are trying to do, but if you simply intend for it to classify positive and negative messages a regular Neural Network might be better suited.

What your RNN does (if implemented correctly) is learn classifications in the context of the sequence. i. e. It is learning: what are the odds of a given input X being positive/negative knowing the preceding N outcomes.

This should already be giving you some insight on why different orderings of the training data skew the results. If your training set is all positives followed by all negatives then the vast majority of your training set will be teaching the network that a sequence of positives almost certainly indicate a positive message and vice versa.

Building a good dataset is by far the most important part of applying machine learning and I suggest you look for some high level explanations of the different types of ML and some general information on bias in data.