This is making no sense whatsoever. I am totally missing something here.
Plugs that come out black and dripping do not melt. Excess fuel, whether it is overly rich or flooding out, which it sounds like at any given time either situation applies to your engine, cools the combustion process. Leaness causes excessive heat that could melt spark plugs.
Have you run this with the air box off so you can see if raw gas is pouring into the intake? If it is, it shouldn't be. The other huge contradiction we have going on is you say you can check the fuel level and it is about where it is supposed to be, but the plugs are dripping, and I am assuming they are dripping gas. Is it gas or is it oil?
I need to back way up here. I realize the bike has 314 miles on it.
1.) are you the originial owner?
2.) has it ever run correctly for you?
3.) If it did run correctly at one time, what has happened to it since it did? Been sitting?
4.) Any recent service work Tune up, routine maintenance or repairs done recently?
5.) Have you done a compression test in the last couple of weeks? If not, would you please do that?
6.) Any possibility the exhaust is clogged/plugged up? Feel for exhaust pressure at the ends of the mufflers just to see if it has reasonable exhaust flow.
I am torn between bailing out because we can't make any headway and nothing makes any sense. Everything contradicts conventional wisdom and sticking it out because I am so fricking curious as to what it is.
If you want me to opt out I will so another expert can try his luck at helping you.