Since you have 2 very different lights on your porch, I susspect the problem is electrical, in which case new lights would not work either, although it's possible the electrician would fix an easy problem while installing new lights.

Yes, we are going to repair the electrical. I’m hoping to get some thoughts on what style of light would look best.

It looks like the house was built sometime in the 20's-30's? Please confirm.

Additionally, I'd remove those shutters as they do nothing for the house but look fake. Shutters should look like they could cover the windows as that was what they were originally designed to do in inclement weather.

how deep is the roof line over the door? you could possibly, maybe more easily have them moved to the ceiling? Otherwise I would do nothing that draws attention to them as to placement. what size is the current left light?