Do you have a heat insulator between the intake manifold and the carbs? If not, you may be experiencing percolation. Heat transfers from the manifold (I'm assuming that since it's an Offy, it's an aluminum piece) and into the carburetors. I don't know much about the Carter Weber carbs, but if they're like Carter/Edelbrock AFB carbs, they're aluminum, too.
Aluminum is one of the best conductors of heat around. When the motor and underhood temps rise, that heat is transferred to the intake manifold. If there's no thermal barrier (wood, composite or phenolic spacer) between the manifold and the base of the carburetor, the heat is transferred to the body of the carburetor where it transforms the float bowl into a percolator. The heat causes the gasoline to boil away...and once it starts it only gets worse as long as heat is still present.
When you shut the motor down, (or it does it for you) the heat can decrease to the point where the float bowls are below the boiling point of gas...and the incoming gas from the fuel pump will cool it even more...until the cycle starts all over again.
I've seen this happen over and over again on smallblock V8s (big blocks aren't immune, tho) where someone has put on an Edelbrock aluminum manifold and an AFB carb that they bought at the FLAPS and as soon as the weather gets hot, they start experiencing problems. By the time they're replaced the carb, the coil, the distributor and a bunch of other stuff, they might take the advice given them and put a heat insulator in and, "voila!" the problem goes away.
I can't say with 100 percent certainty that it's your problem, but I'd say there's a pretty good chance of it.
