actually these are power regulators, and typically they would be hot if you have significant current draw.
Since the screen starts at all (from cold), I think these regulators are just fine, but something takes too much energy so these warm up too much.
The most power consuming circuits in the screen is backlight, I'd check on that first.
Anyway, can you figure overall current draw for whole screen(with multimeter plugged in line with power)? By this you'll find out if the screen is within parameters or not.
I would suggest to measure voltage on output of these 2 regulators. Typical screen needs 2 voltages: 3.3V or 5V for logic, and 12V for backlight. Make sure these voltages are within range of +/- 0.25V
See if any other parts are getting hot. But not with your finger directly on chips/transistors, I still remember my burns on fingers from some processor
