How to implement OpAmp floating input protection
Without the input voltage applied the opamp is not in open loop. It still has feedback but the feedback has become positive instead of negative. I do not think offset has anything to do with this.

All 4 feedback resistors are 10 k so feedback to both inputs (inverting and non-inverting) is 1/2. If you disconnect the input source then the feedback to the non-inverting input changes to 1 so it "wins" when compared to the (still 1/2) feedback of the inverting input.

I disregard the fact that the negative feedback is from one side of the sense resistor and the positive feedback is coming from the other side. But since the voltage across the sense resistor is small I think I can safely do this.

A possible solution might be to apply the input voltage on the inverting input side and ground the left side of Rin1. That way when there is no input voltage source applied the negative feedback would increase (which is OK, I think) but the positive feedback stays at 1/2.

You might also have to swap the feedback signals around the sense resistor since you would be introducing the input voltage at the negative side now.

If possible, compare both situations in a simulator and see if that resolves the issue.