I'm a little confused how you have connected it and how it is powered.
I had never 5V leds flicker when writing to SD card. You have to look into that.
Did you use that format tool in the link ? Did you try other SD cards ?
You may not connect Arduino 5V signals directly to a SD card. Which module do you use ?
Could you measure the 5V pin. It should be 5.0V, if it is 4.5V then it is too low (in my opinion).
Many sensor work at 3.3V. Are they connected to the I2C as well ?
Yes, a Serial.println("Hello") uses ram. Use the 'F()' macro to avoid that.
Serial.println("Hello"); // The text "Hello" is in rom and in ram !
Serial.println(F("Hello")); // The text "Hello" is only in rom
The other module could run at 3.2V and it is connected via I2C ? That makes me frown.
How are the pullup resistors connected ?
Current could flow into the pins of the 3.2V Arduino, or the 3.2V could not be recognized by the 5V Arduino. A level shifter avoid problems and makes it reliable.
Perhaps the I2C bus is working, and you have not enough ram. There could also be bugs in the sketch.
Do you have a schematic and can you show the sketch ?