AC to DC rails sag excessively when loaded
The resistors between your filter capacitors are counterproductive. They cause extra voltage drop due to load current, and prevent the later capacitors from charging to peak rectified voltage. 

Here's what happens to the unregulated output when a load of 100mA is applied (simulated in LTspice):-



Voltage is very smooth but has dropped to 14V, which is right on the dropout voltage of a 7812. 

Your regulators remove the ripple, so the filter capacitors just have keep the unregulated voltage above the dropout voltage of the regulators. The circuit will work better if you simply join all the filter capacitors together to make one larger capacitor.  

Without resistors and the 2200uF capacitors paralleled to make 6600uF the unregulated output looks like this:-



Now the voltage is staying above 16V so the regulator has plenty of headroom. There is a small amount of ripple, but the regulator will remove it.


  The pre-regulator positive rail measures well over 14v (I believe
  almost 15v) and the regulator has a 2v dropout voltage, so I thought
  it'd be okay.


If the transformer is really putting out 12VAC then you should be getting over 16V with no load. Under load it will drop, but should still be good for up to ~100mA. Transformers usually put out higher than rated voltage when unloaded to ensure that they meet their spec at rated load, so in practice it should be even higher. Lower than expected unloaded voltage suggests either incorrect transformer winding or low mains voltage.   

If the unregulated voltage is over 14V under load then the regulator should have enough to work on (at least at currents well below 1A). A meter only reads average voltage so it won't tell you the lower ripple 'trough' voltage, but your circuit should have very low ripple if the capacitors are anywhere near the correct values. So if the loaded regulator input voltage is almost 15V then something else is causing the low output voltage.