the DataSheet shows that the 5v coil has a resistance of 70 ohms and draws about 0.35 watts.
As I understand the output of all the pins is 200mA max.
a bit about relays.
relays, as we use them in the Arduino world, have three power sources.
the contacts are isolated from the coil so they get some power that is not connected to the Arduino.
it could be 220 VAC, and in terms of the DC power supply, as long as it is not connected, we do not care.
the second power is for the COIL. this rated as whatever is printed on the relay. 5volts DC and 12volts DC are the most common, 5volt being the one we use most often. and the one your drawing shows.
the coil is nothing more than a wire that is wrapped. hence the name coil.
in the 5 volt iteration, that manufacture lists 70 ohms as the resistance through the coil and they list 0.35 watts as the power for the coil.
you have to have a power supply that can supply that power.
the THIRD power needed is the power needed to drive the circuit that drives the relay. this is either a transistor or a MOSFET. often a transistor. as you should know the transistor base is connected to ground. so, when you apply power, the base uses lots and lots of power.
to properly drive the transistor, you need to do some calculations. to keep this short, you have to provide a resistor so that the base gets enough power to operate properly.
in many of the coils we buy, the coil is driven with this third power source by means of a opto-isolator.
this is really just an LED in a housing. your power it like any common LED with about 10-20mA and you use your LED calculator to tell you what resistor would work.
in your picture, you show only one power source for both the coil and the driver circuit.
and you do not show a resistor on the signal to the driver circuit.
we often ask that you provide links to your parts and when we don't we make assumptions.
if your relay board has an on-board opto, then it will also have the flyback diode on it.
All that said, you still have not posted your power supply that feeds the Arduino for this project.
further, since you said that your problems start when you add the 12v power, it seems clear that your problems are power source related.