It does sound like the wires are switched, the park light shouldn't be as bright as the turn light. Usually, a solid light on the indicator would mean a burnt out bulb or too much resistance, like perhaps an open circuit.
On the gound, it does ground through the housinf/body, but sometmes if the paint os too thick or there's not enough good metal to metal contact, that will cause problems. I so no problem with installing a ground lead say from the housing mounting studs to a good chassis ground. I had to do that on mine. I ran a pair of ground wires to the same point that the headlamps are grounded to. That helped in my situation (slow flashing).
