No, you shouldn't have power to ground both sides of the flasher unit socket with it removed, only one side should be "hot" with the key "run" position.

With fuses 1 and 5 both "in" and signal/hazard flashers both removed there shouldn't be any power to the directional/hazard switch to power the lights, should be dead.

Only other power source to it would/could be brake light switch but it'd have to be stuck in the "closed" position or someone holding the pedal down. You'd have brake lights all the time, and removing fuse #1 would/should have killed the power via that route anyway.

Looking like you have a fried signal/hazard switch, trying to confirm it.

Both turn indicators in the instrument cluster both work right?