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You can replace sockets yourself easily and cheaply enough, take a photo before you take any wires out of the back for reference if you are unsure.
If the cables inside the socket are all melted then I'd probably get a professional opinion / get the cable replaced.I'd be more concerned about why that didn't trip a fuse somewhere and what caused it in the first place. If you can then check the wiring in the plug itself - though not easy to get those moulded plugs apart. Wouldn't be surprised to see a loose connection somewhere.
So that’s what the weird fish/burning smell I’ve been smelling every time I put the dishwasher on is. Cut/new plug and see a spark for the socket?