How to properly power Arduino Mega with relay block and display?

Question

Arduino

There is an Arduino Mega, which according to the multimeter, naked consumes about 80 mA at a supply voltage of 12V, the backlight of the monitor 16x2 characters from MELT is another 100 mA, the relay is still about 60 mA, that is, about 250 in total. Near the power input, the voltage regulator heats up so, that the hand is hard to hold.
If the board is rated at 5 V, then (12-5) * 0.25 = 1.75 W on a tiny element.
I changed the adapter to 7 with something volts. It heats up weakly. But at the five-volt arduino output instead of 5 V 4.5 V, the contrast on the monitor has also decreased, especially noticeable when the relay is turned on.
How to properly power the board and what current is it designed for?
Update:
Relay: DFRobot Reley module v.3

Answer 1

What stabilizer is on the Arduino board? Is it even capable of delivering 200 mA? The popular L78L05 for such applications cannot provide more than 100 mA.
Is the adapter capable of delivering the required power? How much does its output voltage drop?