I realize this post is over 6 months old, but I am having the exact same problem. My heat grate looks identical. I have tried to clean the grate, but nothing worked until I used a wire brush in a drill to grind it off. It reappears in a day or two. Many clothes have been ruined already. My dryer is a Frigidaire model FAQG7011KW0. It is almost 2 years old. We noticed clothes being damaged about 6 months ago. Before then I can't say for sure whether the black residue was there, but I don't think so. Certainly no clothes were ruined for the first year or so. Any ideas or solutions that have worked would be really great. The dryer vent is not plugged.
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