How does noise affect an unbalanced signal on a balanced cable?
How well this works depends on the output impedance of the source, and the common mode impedance of the input. 

Basically a balanced circuit works perfectly and delivers the full interference rejection benefits even if only one line is driven, providing the other leg is connected to the reference plane at the source via an impedance equal to that of the driven leg. 

Certainly moving the adaptor close to the source is worth a shot, but I have one further question:

Are you sure you don't have an earth loop going on?
Pin 1 problems are still out there, and weirdness in unbalanced gear is fairly common, you might find that moving the adaptor AND disconnecting the screen at one end works best (It makes the screen effective for low frequency electric fields only, but stops circulating current in the ground network). 

A typical XLR cable is at least as well screened as a jack lead, plus the twisting of the pairs gives it MUCH better magnetic field rejection.