First, try starting with the connector, because the odds are about 90% it's the problem. Disconnect it and look more carefully - the green coating can be not just on the outside, but on the actual contacts inside too. If the coating is just surface-level, regular sandpaper will work fine (fine grit, like 120-150) or even an eraser. Polish the contacts until they shine, blow off the dust.
After cleaning, treat the contacts with a protective compound - WD-40 or special contact grease (you can find it at auto shops) will do. This will prevent oxidation from happening again. You'll notice the difference right away: if the lamp was flickering or working poorly before, it should light up properly after treatment.
If the lamp still burns out after cleaning the connector - then yeah, go ahead and check the wiring further down. But in 95% of cases like this, just getting the contacts in shape is enough.