Hi, the black buttons you describe are TOCs (thermal overload cut-outs). It's unusual for them to be resettable. It's useful, but most of them have historically been one-shot devices like a fuse. When they trip, they are gone, and need replacing. The reason for this is that they don't usually trip for no reason. The cause always requires properly investigating. This is exactly what you have found. They have tripped again because something is overheating and needs fixing.
I would also replace both TOC devices too, especially the one that looks overheated.
I have an article on Whitegoodshelp that explains the possible causes for overheating, which include the main dryer fan not running, blockages, and no water running into the condenser - Washer dryer - dryer not heating