No, it isn't you. That model year had plenty of lemons and I got one very much like yours that kept melting itself. When it was time for the third screen replacement, I asked Apple to just replace the computer which they did with a brand new late 2013 which has been trouble free. I had Applecare and this replacement occurred at around 2 years in.
I would request a replacement whether or not you bought Applecare and if you don't get results at the first level, I would ask to be transferred up to the next level and so on until you do. If you cannot get results on the phone, I would write to Tim Cook pointing out that lemons in this model year due to heat issues are a known thing and that your computer was faulty from day one as such, therefore you want a replacement. Do not accept no for an answer.
I guess if Tim said no you'd be out of luck but I don't think that would happen. They know about this.