Opamp Voltage Follower Oscilates
You have exceeded the input common mode range of the amplifier. Read the datasheet. With a 12V supply the inputs should both be between 0V and 10.5V. 

That may not be the cause of your 'oscillation' but it most certainly will not work as drawn. If possible, put the shunt resistor on the low side of the load. You could also consider a rail-to-rail I/O amplifier but even that will not work for inputs right up to the power supply voltage- there has to be a bit of voltage for the amplifier to work.

You are also asking a lot of your differential amplifier. It also has to accept the voltage from the follower (which isn't really doing much for you) and the common mode rejection of the differential amplifier will add directly to the error. In other words, when the 12V changes the common mode voltage at the diff amp changes so that has an outsized effect on the amplification of the relatively small difference across the shunt resistor. Any small error in the resistors will cause a large output offset error which will not be very stable. If you have 1% resistors you might get 20% or 50% error, for example. This issue also goes away if you put the shunt on the low side.