BETTER YET, take the damn bulb out and leave it out. What happens then? Try turn signals and hazards.

CLEARLY we have 12 volts on the right side at all times. That circumvents the hazards, as all six lights are tied as one during hazard operation.
The left signals work because, in turn signal mode, the left and right are separate.

There is power being fed to the right side at all times - so where is it coming from? Pull bulbs on the right until that green light goes out and you have found the culprit. If pulling both front and rear out do not turn the indicator on the dash off, it's in the wiring, probably in the Hazard switch.

Its making sense now.