I have an Ad623 which is a current sense amplifier
It is, as AD puts it and can be seen from the block diagram, an "Instrumentation Amplifier".
Unfortunately, its common mode range does not include the positive rail so it's not suitable to directly connect to the high side shunt, see the DS "Input voltage Range", bottom of page 3.
You could bring down the voltage with a resistive divider, but that will affect noise, offset, increase the needed gain, and, more important, won't solve your problems as the minimum output voltage is 0.2V (for a 10kohm load).
To address this last point, you have the possibility to offset the output to a fixed voltage via AD623 REF pin, and use the same reference for the AIN- of the ADC...but it really seem an unneeded complication.
I had suggested a current sense amplifier as their common mode voltage largely exceeds the supply voltage, but you should find one with the needed bandwidth, transconductance and input range.
But, once again I'll ask: could you move the shunt on the low side and have all of these complications disappear?
Since you are after a 1V range, the measured current will be 2A maximum, it's probably even possible to use a lower valued shunt and higher gain directly in the ADC...