In recent decades, we have witnessed the democratization of AI-powered Educational Technology (AI-EdTech). However, despite the increased accessibility and evolving technological capabilities, its adoption is accompanied by significant challenges, predominantly rooted in social and psychological aspects. At the same time, limited research has been conducted on human factors, especially trust, influencing students' readiness and willingness to adopt AI-EdTech. This study aims to bridge this gap by addressing the multidimensional nature of trust and developing a new instrument for measuring students' perceptions of adopting AI-EdTech. With 665 student responses, we employ Exploratory and Confirmatory Factor Analysis to provide evidence of the instrument's internal validity and identify four key factors influencing students' trust and readiness to adopt AI-EdTech. We then utilize Structural Equations Modeling to explore the causal relationships among these factors, confirming that students' trust in AI-EdTech positively influences AI-EdTech's perceived usefulness both directly and indirectly through AI-readiness. Finally, we use our instrument to analyze 665 student responses, covering eight courses and Bachelor's and Master's degree programs. Our contribution is twofold. First, by introducing the empirically validated instrument, we address the need for more consistent and reliable assessments of trust-related factors in student adoption of AI-EdTech. Second, our findings confirm that student demographics, specifically gender and educational background, significantly correlated with their trust perceptions, emphasizing the importance of addressing the specific needs of students with various demographics.