> the resistance of the coil is 122 ohms.
> I was using a 180 ohm resistor in series
122 ohm load in a 180+122= 302 ohm circuit should get 122/302= 0.4 of the supply voltage. Taking 6.3V it should be 2.54V.
Not clear how you measured 2.7V.
"5V" relays will not reliably flop at 2.5V/2.7V.
> 6.3 volts. I was using a 180 ohm resistor in series which dropped it down to 5 volts
Resistors don't "drop voltage" unless there is a LOAD. Then the drop is largely affected by load.
It appears you got 6.3V through 180r to drop to 5V with only your meter as a load. For that to work out, your meter must be 692 ohms. That's way-way low. VTVMs and DMMs are more like 10,000,000 ohms. Good VOM on 10V range 200,000 ohms. Cheap VOM on 5V range 5,000 ohms.
There is something unexpected going on, either in your method or your tools. More info might clear it up.
BTW, you sure can test "5V" relays with a 6V battery. They do not die at 5.001V. Most have a long-term rating of 150% of nominal voltage. This means you can tap-test with nearly twice the nominal voltage (don't hold 2X voltage more than a few seconds).
> I reversed the polarity of my DC
This is a good point. Many "logic voltage" relays have a built-in diode. When driving from chips and such, a diode may be needed to protect the chip from coil turn-off kick. Adding the diode in the relay is a useful frill for such users. However it does mean that if you connect the coil backward the diode shorts the coil.