It's your Weber - its no good - send it to me and I'll send you a nice Solex. Naw, just kidding.
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You're most likely culprit is the float adjustment or clogged needle valve like Gary suggests.
But, just curious, have you recently installed an aftermarket electric fuel pump?
And, what changes or repairs have you made to the car just prior to your carb beginning to drip?
Does the primary continue to drip after you shut the engine off? and for how long?
Do you still have the heat shield on the manifold?
We've just had some record temps in Northern California, Is your tank too full, with the car sitting out in the sun?
Just for fun, when you notice that primary dripping, open up your secondary butterfly and see if the other barrel is dripping as well.