If what you need is just high CMRR, an instrumentation amplifier may be the best choice. Despite the name, you can use it everywhere and it behaves like a fully differential amplifier (usable also in cases where the common mode input is hundred times bigger than the signal).
The conceptual schematic of that amplifier is this:

$$ V_{OUT} = \left(1 + \dfrac{2 \cdot R_1}{R_{GAIN}}\right) \dfrac{R_3}{R_2} (V_2 - V_1) $$
As you can see, the symmetric structure gives a great performance in rejecting common mode, and the offset of the third op-amp is made less important by the gain in the previous stages.
Most (if not all) the amplifiers have offset correction, which can improved also using an external resistor; they have also programmable gain, also using a resistor.
Here you have a table from Analog Devices where you can choose the proper one.
Update
For measuring currents, you have also the choice of dedicated current sense amplifiers or, even more appropriate, high-side current monitors (like markrages suggested in the comment and here).