There are two things you need to address here: The leaks themselves, and the causes of fuel having occasion to travel through those leaks on engine shutdown.
Nate's suggestion is a good one (double-gasketing the junction between the main body and the throttle body).
You'll also want to check to see that the throttle shaft isn't loose in its bushings-that's another common leak point, and even the less-disreputable carburetor "remanufacturing" factories like Holleys scarcely ever rebush them. One good test for this is to let the engine idle and spray carburetor cleaner at the base of the carburetor where the throttle shaft passes through, on both ends. If the engine idle changes noticeably or if you see the carb cleaner rapidly sucked into the carb at that junction, you've got excessive throttle bushing sideplay.
The necessary rebushing can be done, and warped castings repaired so that they seal, by any of several skilled carburetor rebuilders (not the same as "remanufacturers"). One such goes by the name of the Old Carb Doctor, 800-945-CARB. You mention you've got the original carb. Originals, no matter how grimy and old, are always easier to refurbish to as-new condition than abused "remanufactured" units, so you may want to carry on living with the annoyances of the "reman" unit while you send off the original for a proper refurb.
Mike's babble about the "fuel return line" on your Dodge is his customary hallucinatory babble. There were no Stromberg WW carburetors of any year with fuel return lines. So, there's no such a line to have rusted or otherwise become damaged. His basic idea is sound, though: You need to reduce the effect of percolation, which is when heat transferred to the fuel boils it when the engine is shut off. There are several ways of tackling this problem. A carburetor heat shield can be made in the form of a carb-to-manifold gasket that extends outward several inches in every direction. An extra-thick carb-to-manifold gasket also helps insulate the carb from manifold heat.
Often, percolation begins not in the carburetor but in the fuel line that runs from the fuel pump to the carb. You can rework this line to significantly reduce heat transfer to the fuel: Remove the metal line that runs from the fuel pump to the carburetor. Install a 5/16" IV-flare-to-5/16"-hose-barb brass fitting in the fuel pump outlet and another in the carb inlet. These fittings can be had from any well-stocked hardware store that carries the Dorman "orange drawers" line of springs, fittings, etc. The Dorman part number is 492-024; Everbrass number is 1791.
Between these fittings, run a length of 5/16" I.D. _fuel injection_ hose marked SAE 30R9, not the lower-test old-fashioned 30R7 stuff that doesn't do well with modern gas formulations over time. Be sure to get fuel injection hose clamps, too. Route this line carefully away from sources of heat (exhaust manifold...) and away from moving parts. Make sure to leave enough slack in this hose so that it is not stretched taut at any point.
Think carefully about how you place the fuel filter, and use one with a
*metal* can, because the plastic ones sometimes don't do well with current gas formulations. The fuel filter should be installed in a vertical or near-vertical orientation, inlet at the bottom and outlet at the top, and in as cool a location as possible.
With this setup, a lot of the b=ECtchy hot and cold start problems disappear, because you're no longer boiling fuel in that metal line when you shut off the engine. In extreme cases where it's not possible to isolate the fuel line sufficiently from engine heat, then a vapor-return line such as Mike was trying to talk about can be added. It involves the use of a 3-port fuel filter. The two normal 5/16" ports are present, as well as a smaller 1/4" port near the top of the filter. One such filter is the NAPA #3040. You connect a new 1/4" fuel line hose to that 1/4" port, which must be pointing upward. Loop the 1/4" hose in a full, loose 360=B0 loop, then either run it clear back to the fuel tank where it connects to a new 1/4" nipple you've added to the fuel tank sender plate, or -- preferably -- form and install a hardline for the long length of the truck.
DS