Hello,
both effects are to be expected.
In the first case, the resistors are loaded with 2-25W Watts (I*I*1Ohm), 200mW - 2.5W for every single one.
You do not specify the type , case size and max. power of the resistors, neither the OpAmp type.
If you write Metal Oxide / 5W, then I assume, you are using cheap thick film power resistors, which have a very high T.C. >= 100ppm
Look for wirewound precision power resistors instead.
Better, you reduce the power dissipation by reducing the resistance and the voltage over them by a factor of 10 at least.
There are either power resistors with cooling case, or precision resistors, as thin film, or wire wound with low T.C. of < 10ppm/K or <3ppm/K, respectively.
For first try, ordinary 50ppm/K thin film resistors may be good enough.
The 2nd schematic does not work, will oscillate, because the 100x OpAmp is lacking feedback.
Please, compare your circuit with standard inverting/noninverting circuitry of OpAmps.
In your configuration you need a special "instrumentation amplifier", instead of that ordinary OpAmp.
But you can at first discard the first buffer OpAmp, because the 2nd OpAmp also has low input current on the '+' input.
Then, I would simply use one low offest type or Chopper OpAmp (e.g. OP01, 7650) only.
Divide the input voltage by 10, i.e. 0 .. 0.5V, then directly on the chopper '+', and then use 10 * 1 Ohm metal film (TC 50) or better resistors.
Also, you have to design a 4 point Kelvin arrangement on your PCB, for those 10 resistors, i.e. 2 lines for the current, from the FET to GND, and two lines from the resistors to the '-' of the Chopper, and the GND from the potentiometer/reference.
Frank