Update on the carb weep situation. Over the weekend I bought the sealer I mentioned earlier and applied it to both sides of the gaskets as well as a thin coat on the gasket surfaces. The instructions say to wait until the solvent flashes off before you assemble the components, so that's what I did. It was a little tricky applying the sealer on the carb itself because you don't want to plug up any air passages or leave any big blobs. But I was careful not to do either of those. Slapped it all back together, bolted it to the engine and fired it up.
So far it seems ok. I've run it a few times up to and at operating temperature, and I don't see any puddling where I saw it before. I'm hesitant to say it's all fixed, but I'm happy with the overall results with the sealer. Once I get a chance to drive it, I'll hopefully have a better idea whether it's fixed or not. But for now, I'll call it a win.