Welcome to the forum chef.
"black stringy gunk"? I have no idea what that could be, but no matter what happens, you should try to determine the cause, so it does't happen again. More than likely something got in there, and when you clean it, it'll be over. Replacing the petcock is a good idea, use a couple wraps of teflon tape on the threads. (I am aware that teflon tape is not gasoline resistant, but it does work in this application.) I would suggest not running the engine until you have washed out your fuel tank, replaced fuel filter, replaced all fuel line (using hose clamps), and cleaned your carb. That should cost you less than $20, and be done in about an hour I'm guessing. I'm blown away that it would run like that!
Do some searching on this site for what products people use to flush their tanks.
Hope this helps. Post some pics if you can