Soffit vents ensure a continuous flow of outside air on the underside of your roof.
In cold climates, this prevents two critical issues. First, from condensation on the underside of the roof from moist air in your home or from earlier in the day that has made its way into the attic. And second, from snow that falls on the roof from melting at the top, and refreezing further down creating an ice dam that causes melt water to backup into your roof and ultimately into your home.
In warm climates, the main benefit is to cool the attic space to reduce your heating costs. Some building products break down in extreme heat. Insulation in your ceiling is only slowing the transmission of heat, but when you have a 120°F attic, you'll have a lot more heat transferring through the insulation than if it is only 80°F.
That said, older homes, and those built in warmer climates are less likely to have soffit vents. Between the gable vents and ridge vents, you likely have enough ventilation for your climate (not that more wouldn't be better).
For more details on roof vents, see tester's fantastic blog post.
In case you haven't noticed, weather varies ;) This alone could account for the different effects each winter. But I think you're right, other factors are at play here.
Ventilation. Your gross vent area is 1/100 the attic area, accounting for louvers, let's say about 1/150th of attic area. If you do not consider the gable ends as cross ventilated, it is barely adequate. But as a cross ventilated arrangement, it is more than adequate. Will more help? Yes. Will it be effective? Harder to say. More on this later.
Moisture. Frost on nails alone is not necessarily cause for alarm. Black mold growing where the ventilation is less effective is definitely cause for concern. I'm no mold expert, but you should at least spray some moldicide on it while you devise a more permanent solution. Since the attic is always warmer than outside air, for there to be enough moisture to condense in the attic, the moisture had to come from either inside the house or from a roof or plumbing leak. Running a humidifier frequently would significantly contribute to the amount of moisture migrating into the attic. You don't mention anything about a vapor barrier, you quite likely do not know.The amount of moisture you observe tells me you don't have an effective moisture or vapor barrier. Not only can this cause mold, but it will also diminish the effectiveness of the insulation.
Why frost on only one side? Most likely solar access. The nails are colder on one side because the roof angles are different. This has significant effect on roof surface temperature, even under a good layer of snow.
Not only can vapor migrate through the ceiling, it can be carried in through air leaking around openings in the ceiling, in particular the pull down steps. If the perimeter is not weatherstripped, much moisture (and heat!) can escape this way.
What to do. Your primary focus should be to limit moisture migration into the attic. This will not only solve the condensation problem, but also limit heat loss. Seal all ceiling penetrations. Not only the steps, but plumbing and electrical penetrations as well. Apply an effective vapor barrier. This is not easy. You don't mention your ceiling finishes, this can make it easy or difficult or expensive. You could simply sandwich plastic between the current ceiling and a thin new ceiling finish. If you have an expensive existing finish, you may need to pull up the attic insulation and spray a barrier down on the back surface.
Note that if your household generates a lot of moisture, a good vapor barrier could cause you to have problems with condensation in your ceiling finish. The only solution to this is reduce moisture generation and increase whole house ventilation, ideally through a heat exchanger of some sort.
Once you've reduced the moisture migration, you may not have a mold problem in the attic any more. You may not need any more ventilation at all. The areas near the eaves obviously are not well ventilated, so the addition of soffit vents is the best approach if more is needed. You will need to drill many holes in the separation board to gain free communication with added vents.
If that works out to be too much effort, adding ventilation elsewhere can still benefit the lower areas, though not as well as soffit vents. Anything that increases ventilation will help. More static vents, power vents, ridge vents, turbine vents, they all have their own pros and cons, but they will all increase ventilation. You can do just one measure or a combination of measures. There is no right solution, but there is something that will make more sense than the other options.
Best Answer
This is a subjective question which depends on your prerogative for several things. How long will you stay in the home (or what's your predilection for saving the planet)? Better venting may keep the attic cooler, which may in turn reduce air conditioning costs (financial and ecological). The payoff (in time and pollution savings) isn't clear without a detailed analysis of current conditions and potential improvement.
If you don't have moisture problems now, and you haven't found cooling costs to be unreasonable, I would not make changes at this stage in the home's impressive life. If you feel like the HVAC system struggles in the heat of summer, it may be a wise move.
FYI, attic vents should remain open year-round. Even in winter there's moisture to be released, especially without a modern sealed building envelope. Much more vapor than you realize may be migrating out of the living space. Also, you want most of the heat to be released anyway to prevent ice damming. You might close them somewhat to retain a bit of heat, but only if it's convenient. If it means pushing into an attic and getting insulation all over, don't bother. Leave them open.