Current Sense circuit with digitally programmable gain
simulate this circuit &ndash; Schematic created using CircuitLab
Something like this would improve your dynamic range significantly.
As Dean said in the comments, this sort of switching is normally done with relays. However, there are many devices available with varying on resistance, leakage and voltage compliance available that could replace them. Look on Digikey, take your time and look thoroughly. You can buy opto isolated FET output switches if you look. You might want to look at ADG1411 (or ADGS1412 maybe better, higher current and SPI interface) type devices to see whether they might be good enough for your application, to integrate several switches into one package. Look at other manufacturers' devices as well. The ADGs are quad, logic level drive, +/- 15 V analog supply, 100 mA or 200mA per path (schematic shows two switches in parallel for the high current range with the 100 mA switch, but you could have 4 ranges with the 200 mA type), leakage in the sub and low nA range.
I've shown the low leakage BAS116 across the switches as path protection, should all switches be open at once. However, if you keep the voltage low enough, then most silicon diodes will do. Don't use schottky, much higher leakage than silicon. Interestingly, every time I've measured the leakage on silicon diodes in the 100 to 200 mV drop range, the big beefy 1N400x has been lower leakage than a 1N4148.