One of those big Hudson sixes would be nice... or a big GMC six...

A badly worn distributor shaft/bushing could cause erratic timing. A problem with the mechanical advance mechanism could also cause it. A badly worn carburetor throttle shaft or a loose throttle plate will cause problems. A crack or gasket leak at the intake manifold could cause erratic operation. **There have been instances of intake manifolds rusting or cracking through into the exhaust heat jacket around the intake plenum under the carburetor. This can happen with no external problem being evident. It can cause very erratic engine performance with no evident cause.** Ignition switch problems can cause issues. The small flex wires inside the distributor can chafe and cause intermittent shorts when the breaker plate moves. Once in a while, a distributor may not have a good ground to the engine block. This usually affects Delco units that move in the block when the vacuum advance operates. Many distributors have a small, flexible ground wire from the breaker plate to the distributor body. If it is missing or broken, it can cause erratic operation.
PS: On 37 120 models, the inner fender panel can be removed to access the valve covers and lower radiator hoses. It's much easier than removing the entire fender, and much less likely to result in paint damage. I think the 38s are the same.

I sounds like you definitely have some driveability issues. A number of things could cause the symptoms you're having, and most are NOT difficult to diagnose and remedy. The engine should behave well under all conditions, and it should have adequate power for all normal driving situations, with some to spare. Job One is to determine the CAUSE of the problem(s). The problem could be as simple as a frayed wire, or a leaking vacuum hose, or a bad coil or plug wire(s). A problem with the distributor rotor, for instance, could produce all the symptoms you describe. Diagnosis comes first.

You don't have to remove it. Refer to the shop manual for adjustment procedure.

Make the pitman shaft adjustment with the steering box in the straight ahead position. Some play is normal when the steering box is near the left or right stops. There should be no play with it on center. The worm shaft adjustment can be done at any position, I think.

If you set the pitman shaft adjustment with the steering box OFF center, it will bind when turned left or right.

That looks like dye to me. Could it be some kind of additive product?

It's said that some tire cleaner products may harm whitewalls. Coker Tire has a recommended product for whitewalls. I just put a new set of Firestone 7.00 X 16 whitewalls on my car. If you look twice at them, it will leave a mark. As the tires age, they tend to yellow, and the surface will develop micro-cracks that make cleaning more difficult. The yellowing may be bleed-through from the black rubber underneath the whitewall overlay. Some tubes may contribute to yellowing. From what I've seen, they all do it.

I've seen a few old whitewalls that have laid out in the weather for years that have bleached back to a bright white. The surface is dry and rough, but white as new.

Most all coils for older applications have the primary terminals marked + and -. Be sure to determine the ground polarity of your car, and connect the coil primary as appropriate. The wire from the coil to the distributor should connect to the primary terminal on the coil that corresponds to the car's ground polarity. The coil will work either way, but it will work better if the polarity is correct.

Compressor speed is low at idle, and engine fan speed is also low. Both of these conditions reduce A/C performance, and road draft is also lost. An auxiliary electric fan might be helpful in traffic, if it could move a LOT of air. At idle and low speeds on sunny days, the exterior of the car can get very hot, which further challenges the A/C. Setting the engine idle a little higher than normal will help cool things off at idle if you have a manual transmission. With an automatic, I'd stick with the factory spec on idle speed. Keeping everything as clean as is practical helps heat escape. Radiator, evaporator, and condenser need to be kept free of bugs, grass seed, dust, etc. and keeping the engine and transmission clean helps cooling. Keeping door and window seals in good condition will help, and any floor, firewall, roof, or or other insulation needs to be in place and in good condition.

If the bearings are reasonably tight and the shaft is not scored or corroded, repacking will usually fix the problem. The Literature archive on this website has a lot of service information that will be helpful.