Yes, the ESAI is the better option for the AD1397. The regular SAI can only do one stereo input and one stereo output while the ESAI can have more channels.
Since not all SoC pins are available on the module edge connector, the ESAI_TX2_RX3 is not available on the Apalis iMX6. This means the ESAI can only feature up to 5 transmitter and 3 receiver interface pins. Since the receiver pins are shared with the transmitter pins, this means you can use the 5 transmitter pins only if no receiver pins are used or you can use only 2 transmitter when 3 receiver are used.
In the simple stereo mode (I2S like), each transmitter or receiver pin can transport one stereo signal. In your case, this means for 4 channels, you actually need only 2 data lines. This means, you can use the AD1937 for 4 channel input and 4 channel output by using 2 RX and 2 TX data pins of the ESAI interface.
Besides the two RX and two TX data pins, the AD1937 requires the RX_FS, TX_FS, RX_CLK and TX_CLK, but it does not require the HF_CLK signals.
The Pinout Designer includes the pin 96 for the RX_HF_CLK, but by default it is using pin 72. However pin 72 should not be in the Pinout Designer, since it features a level shifter on the module which makes the RF_HF_CLK function unusable. I have noted this issue in order to correct it in a future version of the database of the Pinout Designer.