How to speed up Due ADC
Yes, there is a possibility to configure ADC to run faster, as it can handle 1Msps. For four ADC channels it means 250ksps each (with 1us time difference between samples). It's not possible with Arduino wrappers as they are way too slow and also waits for conversion end. You have to configure it directly in MCUs registers, so start with SAM3X8 datasheet.

In short: free running mode, sequenced readings like: 0 1 2 3, with interrupt on 4th sample, interrupt handler and some processing.