KubotaSam--good link. That carburetor is pretty close to mine. The rebuild kit I got included a rubber seat, but that area of the carburetor is different--the seat was too small. There was no seat in it when I took it apart, either. The tip of the original needle appears to be rubber. The kit included one like the original and a shorter one without the rubber tip. It also included a replacement for a carb with a removeable seat assembly. It looks like there are several versions of this animal.
I will take it apart one more time and try the suggestions here. After the wife's latest additions to the honeydew list are out of the way.
Thanks all for your ideas.