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Developing problem-solving competency is central to Science, Technology, Engineering, and Mathematics (STEM) education, yet translating this priority into effective approaches to problem-solving instruction and assessment remain a significant challenge. The recent proliferation of generative artificial intelligence (genAI) tools like ChatGPT in higher education introduces new considerations about how these tools can help or hinder students' development of STEM problem-solving competency. Our research examines these considerations by studying how and why college students use genAI tools in their STEM coursework, focusing on their problem-solving support. We surveyed 40 STEM college students from diverse U.S. institutions and 28 STEM faculty to understand instructor perspectives on effective genAI tool use and guidance in STEM courses. Our findings reveal high adoption rates and diverse applications of genAI tools among STEM students. The most common use cases include finding explanations, exploring related topics, summarizing readings, and helping with problem-set questions. The primary motivation for using genAI tools was to save time. Moreover, over half of student participants reported simply inputting problems for AI to generate solutions, potentially bypassing their own problem-solving processes. These findings indicate that despite high adoption rates, students' current approaches to utilizing genAI tools often fall short in enhancing their own STEM problem-solving competencies. The study also explored students' and STEM instructors' perceptions of the benefits and risks associated with using genAI tools in STEM education. Our findings provide insights into how to guide students on appropriate genAI use in STEM courses and how to design genAI-based tools to foster students' problem-solving competency.