I'm having the EXACT same problem.
At startup, the car would puff out black smoke till it got warmed up. I took it to the mechanic for a tune-up and some other work (New water pump, etc...) It ran flawlessly for about 2 days. Then the black smoke came back. A friend of mine smelled it and said "Yup, that's gas" Now I also have a chugging engine. I tried the screwdriver trick to no avail.

One thing I did notice is that when I got it back the vacuum hose was disconnected from the wiper pump and left hanging I hooked it back up and the engine started to chug. So I took it off and it hasn't stopped chugging since; with or without the hose attached.

What now?

Success!
I put the cluster back together last night. I even fixed the clock! I laid it in and put the lights in with a pigtail ground and HOLY SMOKES! This thing looks like a luxury automobile now!
I did do a small modification (May the Packard gods forgive me.)
On the fuel and temp gauges, the old orange arrows were very dingy and hard to see even when the units were out of the cluster in full sunlight. I thought to myself "how can I brighten these up? They are such fine print"
The Solution: I took an X-acto knife and very carefully scraped the paint off over the arrows. The arrows are now bare metal which is silver and have much more contrast. I left the dots on the fuel gauge alone and just did the arrows. Then I hit the gauge faces with some krylon clear so the newly exposed metal wouldn't rust/tarnish/darken.
The cluster really looks like a million bucks!
Pictures coming soon.

ps. yes, I've been using the correct bulbs the whole time.

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Thanks for the info.
I'm assuming the "regulator" in the service bulletin is just a dropping resistor. Do we know how many ohms?

Well Here I am again with the ongoing saga of my instrument cluster repair.... He he a LOT of fun :)

I have a question about the voltage regulator for the temp/gas gauges. I have grounded it with alligator clip lead and am taking a no-load reading between the regulator output and ground with my old trusty Simpson analog meter. The voltage is bouncing between 6v and 0v almost like a turn signal flasher would. I don't know how these old regulators operate. Do they need a load to operate properly or is this one toast? Also, how many amps are pulled through them? Can I pop in a modern 7905 voltage regulator in it's place? They are rated for 1A.

My Temp gauge always seemed to run high (Before and after a full service of the cooling system) And my gas gauge will only operate between 1/4 and 3/4 full. Previous owner just installed a new gas sending unit from Max Merritt before I purchased the car. Could these be symptoms indicative of a bad regulator AND a bad ground going to the cluster?

That's what I've decided to do, and am doing right now.
Everything is getting cleaned and the inside is getting painted bright white.

I have the whole cluster apart for a good cleaning.
One thing that's always bothered me is that the gels over the oil and batt indicators had fallen down when the tape got hot and goey. Instead of being half-covered they'll be all red again.
To me, those little things are worth the effort. I have to look at them every time I start the engine.

As Bigkev and Turbopackman suggested; I do think a lot of it was a bad ground going to the cluster. Once I Had the front bezel off the cluster I put on an alligator clip to pickup ground. The bulbs shone much brighter!
When I reassemble it I'm going to make a pigtail to get a good ground to the cluster.

I'll keep you updated... Speaking of which, I'd better get back in the garage and get back to work! Thanks Guys.

Yeah, I've bypassed the dimmer and they're just as bright.
I think I'm going to line the inside of the cluster with that reflective tape HVAC guys use. I think it's mylar. Should make it more mirror-like, hence more light.

I went to the hardware store and bought a black-light bulb.
I've confirmed that's not the issue.

I really think it's the black paint on the inside-front of the cluster that is absorbing all the light before it gets to the glass.

And I still have no idea what the metal boxes around the bulbs purpose is. Perhaps to make the light more even? Or so the bulbs don't glare in the instrument lenses? Perhaps they were once reflective inside. Today they are just metal shades that block all the light.

I have attached a photo of the "Baffle"
Top-down rear view from where I removed the Speedometer.

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Well, I did end up pulling the speedometer out of the 300 cluster. I'm glad I did. It gave me a chance to lube all the gears and mechanisms inside.

But "While I'm in there" there is another issue I'd like to address; that of dim instrument cluster illumination.
I have replaced all the bulbs with the correct type, cleaned the dimmer rheostat, and still the lights are dim. Now that I have access inside I have a few questions about whether or not what I have is normal.

Inside the cluster the back-side of the chassis is bare metal, and the front is painted black. The strange part is the back is dirty/rusty and the front looks brand spanking new. This leads me to wonder if the front of the cluster chassis was painted black by a previous owner and is not reflecting enough light into the gauge glass.

Also, the bulbs have metal "reflectors" (for lack of a proper term) that surround the bulbs, boxing them in and only allowing light out the top.

From a Google search on this topic, a guy said that Studebakers had black-light filters over the bulbs and he thought Packards might have them also. Which kind-of makes sense as the gear shift indicator is this sort of arrangement on my 300.

So the question is. What's supposed to be in there and how's it all supposed to work?

Thanks Again!