First post, and a few questions

[1BAD68]

Thats a Carter 4401S AVS carb, just Google Carter AVS and find a ton of info or heres an article on rebuilding one...
http://moparmuscle.automotive.com/30336/mopp-0305-carter-avs-carburetor-install/index.html

The "dashpot" is completely different than an accelerator pump. I remember the dashpots were for idle adjustment when coming out of the pedal, somehow it keeps the engine from dying.
I'm not sure that would be your problem from the description you gave but your Grandpa probably knows more than me when it comes to carburator's.
Anyway, when rebuilt and setup properly that is an excellent carb so hang on to it.

Oh, and dont lose that "fender tag" he gave you.

[69DodgeCharger]

You might want to check the metal surfaces of the inside of your distributor cap, the tip of the rotor and also check the condition of your points for corrosion if the car hasn't been upgraded to electronic ignition yet. You can clean them up with some fine emery cloth or a fingernail file. This might help until you replace these items.

I don't know if it was mentioned but you should pull the  spark plugs and check their gap/condition too. Sometimes the diaphram in the fuel pump will go bad from sitting for a long time too. Did you drain all the old fuel out of the gas tank? Change the fuel filter? Put in some fresh gas? Is there rust in the tank that keeps pluggin up the fuel line and filter? Those are a few things  you can check easy enough and eliminate them as causes of your misfiring.

The fender tag goes on the top of the drivers side inner fender between the battery and the outer fender.

[Dans 68]

...I've looked around on the site and on the YearOne site for a diagram or exploded view of the carb, but I can't seem to come up with one....

Here you go. If you need the other three pages let me know and I'll scan them.

Dan

[6T8]

Thanks for all of the help guys. I'll keep you posted.

I swapped out all of the plugs the other night and tested compression on all cylinders, and I think I'm in good shape. I'm sure the carburetor still needs some love but it's running well enough to please me for now. 

Gonna jump into the brakes this weekend and see what I'm up against.

I appreciate everyone pitching in with all of the information.