look closely at the cables that connect the sensors to the coax plug at the inside of the fenderwell. If they are cracked then its likely that is the culprit. If your lucky it might just be one cable and the sensors are OK, the cables take a lot of beating and have to flex with the movement of the wheel so they tend to go bad near the end where it fits on the wheel.
I recommend fixing the ABS its a good design for a purpose.

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'95 E320 Wagon my favorite road car. '99 E300D wolf in sheeps body, '87 300D Sportline suspension, '79 300TD w/ 617.952 engine at 367,750 and counting!