You verified that the temperature readout on the chiller is correct across the range of temperatures in which it operates correct?
That's all it does, it's a sensor, specifically its' resistance value changes with the temperature. Which the microcontroller senses, and the microcontroller is what tells both relays what to do, so actually they have quite a bit more in common (electrically) than just a temperature sensor.
If the readout agrees with your other thermometer no matter what the water temp is, then it is not the temperature sensor. IE if you measure water temp with a thermometer and it says 60, and the readout on the chiller says 60 then the temperature sensor is not malfunctioning. They need to be the same, if the temperature on the thermometer says 65 and the readout on the chiller says 58 then the temperature sensor is suspect.
It's very hard for me to try and explain what you need to do and why, because I haven't had to do that for a long time. Everyone that I work with knows what different electronic components are and how to perform tests correctly, and how to follow procedures. So it's been challenging, but I think I have helped you as much as I can.
There are 5 pins on those relays, two for the coil, NC NO and COM. It looks like the hole for the NC pin is unpopulated so you would have to cut that one.
The voltage is dropping because there's a load on the circuit when both relays are active.