Please post the schematic.
The ULN2003 is not the best device to use in this applicaton because it needs a relatively high input current to saturate the output transistor, you might be better using a CMOS buffer or inverter which needs less water conductivity to operate and also gives a more sudden output change as the input voltage rises and falls.
I would also suspect your have a problem with the flow of current through the two tanks. For example, are they metal tanks and if so, are they electricaly bonded together?
The only way you can reduce the temperature of the regulator is to fit a heat sink to it. The power it dissipates is (Vin - Vout)/Iout or about 7 Watts per amp. There is a possible way around your problem, if you use a CMOS buffer to sense the water level and use it to drive a ULN2003 you might be able to remove the regulator altogether and run the LEDs from the 12V line instead of the 5V line. 4000 series CMOS will work up to 15V supply so you might not need the regulator, 74C series will neeed the regulator but moving the LED power to the 12V rail means the LED curent doesn't flow through the regulator and it should run cooler.
Brian.