Determined that the Teensy is unable to output a 40kHz sine wave from the DAC port. Instead of a clean sine wave, it was a choppy square-ish wave. Upon further research, it seems that the DAC on the Teensy is only capable of about a 5kHz sine.
Further ran tests with the ultrasonic transducers using a function generator and oscilloscope. We found out that we don't actually need a sine wave to drive the transducers, and that a square wave would work as well. This is good news since square waves are much easier to produce than sine waves.
Tested transmitter with 3.3Vpp, 40kHz square wave. Result is 400mVpp sine wave. This isn't bad, (much better result than previously), but we would need an amplifier circuit to increase this signal.
Ran some more tests with the current program.
Tested the differences between successive measurements between a single mic and they seem pretty consistent (minus the pop in the beginning). When averaging over a large amount of samples, the result is very good.
Attempted to measure the amount of time that each interrupt spends inside their routine by by saving the difference between the clock cycle entering the interrupt and the clock cycle when leaving the interrupt into an array. From some tests, it seems like they're both consistent, but mic2 is a bit funny. It may be because I'm reattaching mic2's interrupt after mic1, but mic1 is solid.
Instead of implementing the error-filtering code directly on the Teensy, I took some raw measurements and am going to try to filter them out using other methods first, to see if the results are good.
Measure the clock cycles involved with outputting a 40kHz sine wave from the DAC. If it's not too much, it may be feasible to drive it from the Teensy.
One possible alternative is to use thresholding to detect a wave, and then run some ADCs to determine the phase shift afterwards.