Dark chocolate is actually more heat resistant than other kinds. Milk chocolate can scorch at 115° F / 46° C or higher; semi-sweet can have problems at 125° F / 52° C or so. It's mainly the dark chocolate that can withstand temperatures as high as 140° F / 60° C.
I think there are two misconceptions to clear up here, those being:
Baking at 350° F does not mean you are raising the internal temperature to anywhere even close to 350° F. Even yeast breads are not baked to an internal temperature higher than around 200° F, and it's much lower for soft breads and especially cookies. It's hard to find much data on the recommended internal temperature for cookies, but I'd estimate it to be around 160° F simply for food safety reasons.
Just because a food can burn above a certain temperature, does not mean that it will happen immediately. Just as heating oil slightly above its smoke point does not immediately result in flames and rancid taste, heating chocolate above the aforementioned temperatures will not immediately cause it to scorch. What it means is that chocolate can be sustained in a melted state indefinitely at lower temperatures, but raising the temperature further will cause it to eventually scorch. The higher the temperature, the less time it will take.
These "scorching" temperatures are mainly cautions against direct stovetop or microwave heating; with these methods it is easy to get the temperature very high, very fast. When you bake cookies, you are applying very slow, gradual heat; if the temperature of the chips even gets as high as 140° F, it won't stay that high for very long. It's not enough to burn, and that is why even milk chocolate or white chocolate chips tend to do fine in cookies.
So just bake them, and don't worry. The cookies themselves will probably scorch before the chocolate does.
P.S. As SAJ14SAJ writes in the comments, this assumes that you bake the cookies on some kind of insulating material like parchment or a baking mat. There is a risk of burning if you bake them directly on a metal tray and the chocolate comes into direct contact with it, as it will then heat up very quickly to approximately the same temperature as the tray. So don't do that - if you use a metal tray or cookie sheet, make sure there's something between that and your cookies. They bake from the ambient heat in the oven, and don't really need direct heat from the tray.
You can, but I cannot think of a single good reason why you would.
325 F and 350 F are not that far apart; they may even be within the accuracy of most ovens, especially older ones.
When you bake the cake, the two main modalities of heat transfer (radiation and conduction from the air in contact with the surface of the cake) begin to heat up the surface of the cake at the cake/air or cake/pan interface.
Within the body of the batter, conduction begins to transfer heat into the center of the cake.
At the same time, on the top surface especially, water can evaporate, faster than it is replenished from water diffusing from the interior of the batter. This allows crust formation to begin, with drying of the surface, and heating of it to levels above 250 F or so which allow caramelization and Maillard reactions to begin in the crust.
The interior of the cake can never exceed the boiling point of water, and so browning does not happen.
Thus, the higher the temperature, the greater the difference between the surface temperature of the cake as baking proceeds, and the interior of the cake. This has several consequences:
- Thicker cakes should be baked at a lower temperature to permit them to cook through before the surface is over-baked; similarly, thinner cakes can be cooked at a higher temperature
- The higher the temperature, the more rapidly crust formation proceeds, and the more slowly the center of the cake bakes
However, 25 F is not enough of a difference to worry about in most cases.
Starting at only a slightly higher temperature, when the water from the surface is not yet evaporated, will have only a minimal effect on any of this. There is little, if any point to doing so.
Best Answer
If you used a reputable recipe, followed it without variation, including using the same size pan as was specified, and adjusted for altitude, if appropriate, then:
The problem was likely with the thermostat calibration on your oven. The best thing to do is get an oven thermometer and use it to find out how much you need to adjust your temperature control. This answer to a question similar to yours may help with that.
If you can't do that right now - adjusting the temperature up in small increments (10-25 degrees F) and noting what happens, until you get the desired results, is a good plan. Remember that there will always be variables that can affect the cooking time and that most times given are approximate.