Alot of interest by old car enthusiasts about the old vs new gas. When their cars don't run they often rationalize and blame the modern gas. lack of lead, or other additives, etc. Gee, my old LaSalle ran great...
I don't worry about the past but look to the future, something I can influence.
Here's one side of the EPA ethanol regs. From what I understand currently, cellulosic can't happen because there's insufficient US capacity and EPA has curtailed their numbers.
Decadal general trend, vapor pressure of gas has been reduced to reduce "smog" producing hydrocarbon emissions from evaporation. Not increased as most suspect. They used low boiling hydrocarbons in the past to control cold start and make winter blends. Of course, the carcinogen BENZENE was in the first gasoline. It's still in there just much less.
Ethanol was introduced during Clinton EPA when someone had a leak in an UST in CA of MBTE blended gas, which was the oxygenate standard at that time. Oxygenates(MTBE) in gas came first after lead was phased out to promote "cleaner" burning - whatever that is - less CO. MTBE extracts with water and polluted some drinking wells. Ethanol will do the same thing, but ADM won the political battle over the "bad" oil companies. But hey, it's much better to get drunk drinking the polluted water isn't it?
Have fun!
http://www.api.org/~/media/Files/Policy/Fuels-and-Renewables/NERA_FINAL_API_RFS2_July27.pdf

Not only an operational problem but a big fire hazard made worse if you store inside.
It shouldn't do that even with modern fuel, even if you're getting E15 or worse. So don't excuse the problem on the fuel, that's a common myth, but does not solve the problem.
Could be a vacuum leak, that's aspirating the fuel or going to the wrong place. Vacuum lowers Boiling Point. Takes a while for the vacuum to dissipate on shut off. Carb might be leaking down through the bore then vaporizing from the manifold heat and vapor bubbling back up.
It would be commonly what sounds like what O_D said, don't assume, check that's all in order underneath. Float could be stuck, dirt in the carb, vent plugged, etc. I'd tear down and rebuild carb first.
I would not run it like that and make sure the insurance is paid up.

Looks like part of the inner fender well is missing, not shown in that picture. The gas nozzle shroud attaches to it and has a bottom lip to attach the trunk pan. You should not be able to see the ground for there. It's all enclosed, that's why they had to cut it out to gain access.

I reported this before, but there's a new product out there for painting on a plated chrome finish. It uses water after the "paint" is applied. I hadn't seen it yet, but was watching "mobsteel" and they were working on a '53 Pack and they showed and described "painting" the bumpers and they looked excellent. Certainly an acceptable alternative to the price and time of plating, if you're in the driver category or low budget restoration. I need to research this more.

Typical beltline chrome strip pieces. Usually the frontmost piece. A Packard styling "trademark" thing like the hood ornaments.

Pix:
https://www.google.com/search?q=1951+packard&tbm=isch&imgil=nIz8hzeW-mes-M%253A%253B4Gq-bYJHDEDRAM%253Bhttp%25253A%25252F%25252Fwww.conceptcarz.com%25252Fvehicle%25252Fz10515%25252FPackard-Patrician-400.aspx&source=iu&pf=m&fir=nIz8hzeW-mes-M%253A%252C4Gq-bYJHDEDRAM%252C_&biw=1061&bih=641&usg=__cF-innnaN7l5bJfvU_UvX7MMfio%3D&ved=0CCgQyjdqFQoTCN3ot6Si1scCFQaqgAodixQNrA&ei=5dPlVZ3iM4bUggSLqbTgCg#imgrc=nIz8hzeW-mes-M%3A&usg=__cF-innnaN7l5bJfvU_UvX7MMfio%3D

Depends on what you want your car to be. If you want the all original as built cylinders and yours are pitted or worn past sealing, you have little choice but to sleeve. The cost is about the same or more of a repro wheel cylinder.
If you just want drum brakes, then go for the repros, because it'll save you weeks of time. But I found some of these had metric bleeders. You can retap and use SAE but risk leakage.
Apple Hydraulics and White post restoration are two common vintage brake cylinder rebuilders. They'll be able to tell you if it needs sleeved or not once they receive the parts. I cleaned up and put the original thicker wagner rubbers on the repros to make them undistinguishable. I had some strange corrosion from wicking on the lid of the master{Kanter}. {Probably bogus chinese gasket material is used.} You can find correct gasket and lids.
If it's driven alot then consider upgrade to dual master and maybe disc conversion. But disc often means can't use the stock wheels.
I tried to find modern more easily obtainable parts to fit but had no luck with fitment but I was redoing a '49 at that time. The bore sizes and hole patterns are typically different, but could be converted if you know what you're doing but may not resemble stock.
Consider as part of your brake rebuild to replace the brake lines and hoses as you'll contaminate the new parts/fluid with rust, etc.

All I have are 4 Phillips 6006 new in the box.

Is that the round 2 filament(hi-lo beam) you're after?
It's late now, I'll check tomorrow if I have a NOS set.
I bought a bunch of NOS headlamps at auction, and I know several fit the P, halogen and non.

FYI:Fronts usually bolt in on framed cars, rears are usually spot welded.
Patching might be the expedient approach depending how badly rotted. The parts you get may well need patched as well, so you're no further along. They'll have to be prepped anyway.
Take it back to good metal and see how many ft2 need replaced vs how much is left. If less than half, patch it. You need one good side to match the lines. If both are completely rotted on the bottom, you'll have no reference to match, except from another car. Plan on cutting out ALL the rust spots. If you don't and don't seal properly, they'll just pop back through over time and ruin your new paint job.
If you've never replaced a fender or whole quarter before, don't. Take it to a competent body shop. You'll quickly be in over your head and have two parts cars...
A bit of 18ga cold rolled, MIG or TIG, and some hand tools will go a long way but need experience to come out well...
Too many butchered DIY "restored" vehicles running around. Bondo is not panel adhesive and duct tape is not sealer. Metal screws and pop rivets aren't factory seams. Most of these auto shows make it look easy and don't do it right and only show some steps to sell some product.