FFT: low magnitude peaks centered exactly 1 Hz apart across whole spectrum
Is this normal?

No

Like a normal artifact of the fft?

No

Possibly something wrong with my data?

Yes. Could be a data problem, could be a hardware problem, could be a SW problem, could be real. You seem to have a very large DC offset in your data. Without knowing what exactly your data, is I can't tell whether that's expected or if it's a problem.