How do I catch edge cases and integration failures in my tests?

You catch the normal paths now. What is left are the strange corners and the places where two working parts break each other. Those slip past everyone, because they live where no single person was looking. Missing them is not sloppiness, it is the hardest ground in testing. Edge cases hide at the boundaries, empty, huge, zero, the maximum, the unexpected type, the thing that arrives out of order. Make yourself a short list of these boundary questions and ask them of every feature, out of habit rather than inspiration. For integration failures, the bug is usually in the handoff, so test the seams where parts meet, not just the parts themselves. After the next bug that escapes you, do not just fix it, ask what kind it was and add that question to your list. Your test design gets sharper one escaped bug at a time. You are not failing, you are building a map of where trouble hides.