Most of the fully differential amplifiers are going to be limited to about 1V away from the rails due to the architecture of the Vocm stage. See page 4 of the following app note (Fully Differential Amplifiers - sloa054).
One problem that you would run into with setting the output common-mode voltage of the amplifier to 0.6V is that each single ended output will only be able to swing 1.2Vpp before it starts clipping or 2.4Vpp differentially centered at 0V. Typically the FSR range of an ADC is set by Vref which is tied to the Vocm pin of the driving amplifier. For a 3.3V ADC you could be limiting the dynamic range and accuracy by setting Vref so low.
What is the full scale range of the ADC?
If you have a +3.3V supply you can use a - 1.8V supply to get 5.1V total and many 5V differential amplifiers could then support the 0.6V common mode.
If DC coupling is not required you could consider using a 5V supply and using AC coupling capacitors and, again many of our 5V amplifiers would work here.
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