The structure group of TX⊕T∗X is SO(n,n). Since w2(TX)=w2(T∗X), this bundle always admits a spin structure and we lift to Spin(n,n). The sections of the associated spinor bundle are not spinors in the usual way but bispinors like you guessed (they have two spinor indices, one from TX and one from T∗X).
A B-field (in the sense of http://arxiv.org/pdf/math/0401221v1 ) determines an extension
T∗X→E→TX.
By the splitting principle, this bundle also admits a sort of bispinor.
Now it is a bit confusing to me since in physical applications (eg. the wonderful thesis of Yi Li: http://thesis.library.caltech.edu/2098/) only ordinary spinors are used, though I think that the (complex) bispinors are ψ+⊗ψ− and ˉψ+⊗ˉψ−. We have Spin(n,n)=Spin(n)×\ZZ/2Spin(n) and accordingly the spinor representation splits as a tensor product. A spin structure on TX I think gives a basis of the sections of the bispinor bundle of simple tensors, so we get well-defined ψ+ and ψ−.