Question, should I use resistors or any other components between the separate DC power supply and the relay modules?
Just directly connect as shown. If rated for 2A or higher, you could use it to power two 8-relay modules.
I did not use a separate power supply for the relay module as I was given the impression that all the MCU would need to source is the current needed to power the LEDs in the opto.
True, but with one power supply you need a GND connection from the relay board to the MCU board. With a GND connection between the boards, you loose opto isolation. With loss of opto isolation, the negative effects of switching AC inductive loads, and contact arcing can get introduced on the MCU board through the GND line.
In any case, I recommend configuring all unused inputs with INPUT_PULLUP to help prevent high impedance pins acting as antennae.
Keep all AC circuitry and wiring away from the DC wiring and components. Also, increasing the distance (if possible) from the MCU board to the relay board is helpful to reduce effects of interference.