OK this is getting strange. I'm not following what you say about the continuity tests, but it suggests something like a pin connected to ground isn't well connected, or some other soldering error.
Do these tests and report back.
Test continuity between "GND" pin on J2 and other ground points (middle pin of 7809 and 7805)
Test continuity between "+9V" pin of J20 and other 9V points (JP, 7809, anode of D4)
Test continuity between "+5V" pin on J2 and other 5V points, just for completeness.
Check D4 and D5 diodes are oriented correctly.
If the 7809 and 7805 middle pins aren't connected to ground properly, that will cause problems. Continuity test (or test resistance) between those pins and a known good unsoldered ground point like J2
IF that passes, then while powered on:
test voltages between J2 GND and the orange dots, check they're all the same voltage, report them.
test voltages between 7809 middle pin and the orange dots, check they're all the same voltage, report them.
test voltages between J2 GND and the four blue dots, check they're all the same voltage, report them.
test voltages between 7809 middle pin and the four blue dots, check they're all the same voltage, report them.
then test voltage at the two other points between the 7809 and the 7805... i.e. cathode end of D4 (where stripe is), cathode end of D5 (where stripe is).
Take a photo of the right side of the PCB (top and bottom)... probably won't do much but you never know... I might see something.