The adoption of artificial intelligence (AI) tools, specifically those based on large language models (LLMs), has quickly led to turbulent changes in software engineering practice. However, despite their potential, AI-based tools are often thrust into the development process with little consideration towards establishing trust, a key factor in the adoption of new tooling. With this work, we address this trust gap by focusing on the needs of software engineers to facilitate a trust-based, synergistic adoption framework in AI-assisted software engineering (AI4SE). We aim to identify the key factors influencing trust in AI4SE and to design a practical adoption framework that covers the concrete steps and best practices for a trust-based integration of AI4SE. Finally, we evaluate the designed framework by applying it to a real-world use case with a professional software development team. By treating trust as a core design goal, this research provides practical guidance to improve the developer experience and facilitate a seamless adoption of AI4SE tooling.