I can't help thinking you're barking up the wrong tree with all this talk of PAT testing (or rather II&TEE). Measuring the current consumption of the appliance when it's working flat out may tell you something, but I think cogitating over the permissible amounts of leakage current is somewhat fanciful.
It's fair to say the damage to the plug was caused by overheating, which could be caused by overcurrent or a high resistance joint.
I've seen this before, but it was with a commercial washing machine in a hotel laundry being plugged into a 13A socket. The fuse would blow every now and again, but the people who worked in the laundry seemed to have accepted a fuse as a consumable, like a light bulb, which needs changing every now and again. Both sides of the double socket (metal clad) displayed signs of overheating, and they'd run the flex across the floor under a mat so they could use another 13A socket. After checking the power consumption, flex size etc I changed the plug and socket for 16A with a breaker in a separate enclosure and it seems to have been fine since.
This was a commercial washing machine though - I wouldn't expect to have to do this with a domestic appliance, least of all one which comes fitted with a moulded 13A plug.
If there were an overcurrent the fuse would have blown (which you haven't mentioned), which leaves a high resistance joint.
Most electricians will have gone to test a ring final and come across a socket with an abnormally high resistance, plugged the socket adaptor in a few times then found it gives a sensible result. This is because the friction from moving the adaptor has worn away any oxidation on the contacts and it now makes better contact.
Did I read that you plugged the drier in when you bought it and it's been like that ever since? It could be that in an environment where they may be hot, damp air coming out of the drier a layer of corrosion has built up between the socket and plug pin, causing a high resistance joint; in the absence of any further investigation I'd be prepared to put it down to this.