Automatic Essay Scoring (AES) is a well-established educational pursuit that employs machine learning to evaluate student-authored essays. While much effort has been made in this area, current research primarily focuses on either (i) boosting the predictive accuracy of an AES model for a specific prompt (i.e., developing prompt-specific models), which often heavily relies on the use of the labeled data from the same target prompt; or (ii) assessing the applicability of AES models developed on non-target prompts to the intended target prompt (i.e., developing the AES models in a cross-prompt setting). Given the inherent bias in machine learning and its potential impact on marginalized groups, it is imperative to investigate whether such bias exists in current AES methods and, if identified, how it intervenes with an AES model's accuracy and generalizability. Thus, our study aimed to uncover the intricate relationship between an AES model's accuracy, fairness, and generalizability, contributing practical insights for developing effective AES models in real-world education. To this end, we meticulously selected nine prominent AES methods and evaluated their performance using seven metrics on an open-sourced dataset, which contains over 25,000 essays and various demographic information about students such as gender, English language learner status, and economic status. Through extensive evaluations, we demonstrated that: (1) prompt-specific models tend to outperform their cross-prompt counterparts in terms of predictive accuracy; (2) prompt-specific models frequently exhibit a greater bias towards students of different economic statuses compared to cross-prompt models; (3) in the pursuit of generalizability, traditional machine learning models coupled with carefully engineered features hold greater potential for achieving both high accuracy and fairness than complex neural network models.