If your outbuilding was wired properly and you lost the neutral, the welder would still work, and so would the 110 volt fan in it. An old Lincoln tombstone welder was wired to use both hot leads and the appliance ground. The fan was wired in a way which is now illegal---using one hot and the ground---but was standard practice for 110 volt accessories in appliances like welders and kitchen ranges at one time. I suspect that your building is wired so that the ground and the neutral are the same line, you lost one hot line, and the weakly glowing lights and welder fan are getting voltage through the neutral/ground line but don't have an adequate ground.
I mean, as long as we're guessing.