Single Resistor on Common Cathode Display Is Not Limiting Current (With 1 segment lit)
If you used the multimeter in current measure mode, and measured over the resistor, that is wrong. You short circuited or bypassed the resistor with multimeter so it prevented the resistor ti limit the current, and too much current passed via the LED and multimeter. Correct way to measure is the multimeter in voltage mode and measure voltage over the resistor. Or current mode when multimeter is in series with the resistor.