How do I write a cover letter that isn't just my resume?

If your cover letters just restate your resume, you are not alone; it is the most common trap there is. The resume lists what you did. The reader already has it. So a letter that repeats it gives them nothing new and quietly wastes the one chance to sound like a person. A cover letter answers a different question than the resume: not what you have done, but why it matters here and who you are. Pick one thing you actually care about in this work and say it plainly. One real sentence about why you want this beats a paragraph of accomplishments they already read. Before your next letter, write one honest line finishing the thought that you want this job because of something specific, and build from there. Start with something true instead of something impressive. That is the difference between a letter they skim and one that makes them curious about you.