The research reported here addresses the use of Game-Based Learning (GBL) and co-design methods in the social adjustment of newcomer children (the children of recent migrant families). Educational games have been shown to be effective ways of learning for adults and children. While the subject of newcomer children has been studied broadly, there are still many open research questions, including the role of emerging technology (such as GBL) in the context of newcomer children's social adjustment. Also, the literature lacks a well-established digital technology framework in the context of newcomer children's social adjustment and the lack of studies in the role of newcomer children in designing such digital solutions. To bridge these gaps, we propose a social adjustment framework customized for newcomer children. It offers an interdisciplinary theoretical foundation, design guidelines, and a procedural model that consists of three elements (game-based learning, cultural inclusion, and co-design). Our research approach focuses on social adjustment problems faced by newcomer children to Canada aged 9-12 who are from Arabic-speaking countries, evaluating a solution using game-based learning, and engaging the children in co-designing such a solution. Our research has three main contributions: 1) it is the first study of the proper use of game-based learning in the context of newcomer children's social adjustment; 2) it introduces and implements a co-design approach to work with and for newcomer children; 3) it provides a customized educational game-based learning social adjustment framework including guidelines for cultural inclusion to help newcomer children in their social adjustment journey. Our first study showed the most pressing social adjustment problems through a fundamental survey study with newcomer children aged 9-12, parents, and teachers. Our second study showed the effectiveness of game-based learning through a proof-of-concept game we created called the New Beginning. Finally, our third study revealed that newcomer children could contribute effectively as co-designer for creating social adjustment computer games for newcomer children. Also, we present educational game design guidelines to work with and for newcomer children, and we conclude with a reflection on re-designing the initial study game and designing a new game called "Together-WeCan."