I guess I would start by disconnecting 3v3 of the teensy to the audioboard bit power the audioboard from a separate 3v3 regulator.

The audioboard is designed to connect directly to the teensy, but keep the 3v3 power pin out and use a low noise LDO with recommended capacitors and see houw much that helps. I was one of the biggest noise sources in a ultrasound circuit we made.