Ordered a kit for ~$20.  It's supposed to contain all the gaskets, jets, etc.  

That "screw" isn't an adjustment, and I did clean it out.  Also the cylinder shown in this picture has a smaller cylinder inside of it which does move, but only about .100" or so.

 When I drained it it was all green fluid.  I'll pull the pin after I find a rebuild kit.  Arrgh!

I bet that bowl was really full of shit.  If this one is like any other small carb I've had apart, I would slide that pin out and remove the float, pull the needle valve, possibly the seat if its removable, and really clean everything.  Usually the kits come wiht a new valve (and half the time its even correct too).  that screw looking thing in the middle you shoudl be able to unscrew and clean, i believe the other end should have a bunch of holes and will be what mists/injects the fuel.  That might be pretty dirty as well from sitting.  Just be mindful of how many turns it took to remove, etc. and make sure its not an adjusment as well or something in your book.  Be sure to run the 93 octane at minimum too, seriously.  The 87 has too much ethanol now and is hell on small engine carbs with deposits and such.