Hope you get the 044 going. Did you check the fuel line for a crack/hole, it doesn't take much(you probably know that though). Let us know what you find.
I did not think it was the fuel line cause a while back I installed one of them tough green ones that is ethanol safe, so I changed the fuel filter ... no change. So I installed a new carb (yeah, I actually had one not in use) and it was PROBLEM SOLVED!!! I'll see how strong it is with it, and determine if I want to leave it on or rebuild the original one!
I'm just glad I figured out what it was as I really think of that saw as my "old reliable". Bought it new in 12/92!
It starts, idles, and revs fine, but I have not put it in wood yet.