Ok, so focussing on one thing at a time; if your 78L09 is hot, then there are really only a few things that could be wrong.
1. You're not supplying 13.8V, and the voltage is higher, or is reversed (easy to check with the multimeter)
2. The 7809 has been soldered in backwards or a pin is shorted or something (check with the continuity mode, unpowered)
3. One or more of the resistors that are connected to pin 3 of the 78L09 (as shown in the diagram)
are the wrong value
Can you measure the resistors with your multimeter and confirm the values? (or check the
color codes, but sometimes they are not easy to read).
You need to unsolder one wire of the resistors, that is easy if you use a small screwdriver to gently lift one
end of the resistor while heating the wire with the soldering iron.
Also, can you take a detailed color photo of both sides of your board, and post it? In case we can read and
confirm the color codes?
EDIT: by they way, I just noticed you said:
When I unsloldered the 10 ohms resistor that is connected to the output of the 78L09, the 78L09 stopped getting hot..
However, the circuit diagram doesn't have a 10 ohm resistor connected to pin 3 of the 78L09. So the board seems to be
different from the circuit diagram.
Can you scan or take a photo of the instructions too, as well as the photo of your soldered board, both sides?