On my 2003 toyota echo. The directional signal on the right side is faulty.
New bulb. The right rear turn signal does not come on. With the headlights
On the bulb lights, but does not flicker. The sockets are interchangeable. So I tried the
Left socket. Still did not work.

Right side front directional signal flicker(weakly) but with the headlights on
The bulb does not come on at all. Have not replaced this bulb.

I guess there is a short somewhere

Did you inspect the car side of the harness for damage or breaks?
Particularly the W-B wire. Follow that to it's ground point and insure it's not loose/broken/corroded.