You asked for suggestions. Perhaps it would be better if you first understood the source of the issue you've described?
Your issue is caused by inadequate load regulation in your power supply. Here's a short video that explains what's happening, and shows one way to test a power supply's load regulation performance.
Your issue is exacerbated by two ...
This has NOTHING to do with the Pi or GPIO.
These lousy modules are unsuitable for use with the Pi (they are poor for Arduino).
They can only be turned off by putting 5V on the inputs (or by setting GPIO to INPUT)!
Using with the PI risks damage!
NOTE there are hundreds of similar Questions by others who have problems with these.
See Can you use a 5V Relay ...
All the GPIO are pulled high or low at power-up. GPIO 0-8 are pulled high to 3V3 the rest low to ground.
The pulls are quite weak (about 50k). If you need a different default you need to add external pulls.
See page 102 of datasheet for the default pulls.
See pinout for the mapping of pins to GPIO.