If you had said this in your first post we'd have moved a lot faster. You just told me that the entire right side is powered all the time, not just the dash light. Makes perfect sense now.
The right side is powered all the time. When you said it was "the light on the dash" you did NOT make it clear that the outside signals were also on (the one, anyway). This made it look like an instrument cluster issue.

But now that we know the rigth side is powered at all times, it all makes sense. When you select left, it works. When you turn on hazards, the power on the right circuit back-biases the flasher, because at that time, left and right are bridged.

Your job now is to find out what is powering the right side, and I'd bet the 'new bulb' has something to do with it. Yank that bulb out and see what you get. If the condition stays the same, we have more work to do, if yanking that bulb turns the light out and makes the four ways work, then its the bulb or socket.