The later Walbro WT-323 carburetors for 009 and 010 had only the low speed screw (think for emissions reasons), but if yours has a brass plug in place of the high speed screw it may be someone messed with the carb. Like lost the high speed screw and just plugged the hole. With the plug there would no fuel flow control at high speed. It would be like the high speed screw was turned out all the way and the saw would run really rich like you say, even puke gas like you say.
There are used rebuildable Walbro carbs for 009L, 011 and 012's on the giant auction site. Almost all the used Walbro carbs there are WT-29a's. The 009 and 010 WT-323 carb had a smaller air inlet than the WT-29A carb on the 009L, 011 and 012. You might be able to use a WT-29A carb on a 009 with a higher idle speed. Would suggest avoiding Zama carb for these saws.
I have only one 009, and about 7 40.8cc 009L's (with 2 having the 40mm cylinders for 45cc displacement, lots of fun)
Stihl 024 to 048 even numbers, 08s's, S10's, TS350, and the 009L's
Solo 662, 652, 651, 616, 611