The polarization going into the EOM was off by 90 degrees. This must have happened when I was minimizing the DC offset at some point (since there is a minimum whenever we are aligned with either of the crystal's axes, but of course the phase modulation is minimal here). I fixed it and we now have reasonable sidebands:
The carrier peak is at ~11 V, the sidebands are at ~176 mV. Using the 1st order bessel approximation, the modulation depth is
Γ = 2 sqrt(0.176/11) = 0.25 rads this is about what we want.