this is a bit of a long winded story/fix, for answer, skip to last few paragraphs.
I had a similar problem a few months ago. I had bought a pair of 9006 (low-beams) Sylvania Sliverstar Ultras a few years back and then after a few years one of them went out (they would flicker sporadically, bumps on the road would wiggle the connections). I pulled the bulb out and the connector looked black, singed/burned/melted, the bulb end looked ok though, singed but not melted. I ended up cutting the pigtail off and putting a junkyard one in and put the same (singed) bulb back in.
A few weeks later the same thing happened to the other side. I bought a brand new pigtail from Advanced Auto and soldered that in. I think, I don't remember exactly, I went through maybe 2 or three more pigtails (they kept burning up/melting) before smacking myself on the forehead for being a cheapass/dumbass, I kept putting in new pigtails but putting the singed bulbs back in!
I guess that when the bulbs were at the end of their life (Silverstars have higher than oem light output but shorter lifespan), instead of the filament winding burning and going Open, the first thing to go was the connector, that's why they were melting (I'm also going to venture a guess and say that the sylvania bulb's plastic had a slightly higher melting temp than the pigtail connectors, it would seem to make sense).
I figured that the singed/blackened stuff on the bulb leads were creating a high resistance which caused the connectors to get excessively hot and melt (since I kept using old, singed bulbs I ended up going through a few too many pigtails).
long story short, get both pigtails replaced with new ones and put in oem or equivalent bulbs, not those brighter supposedly "better output" bulbs. I now just use plain old GE bulbs from Walmart, cheap and effective. I do still use the silverstar ultras for high-beams since their lifespan should be longer since they are not always on.
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