The rear running lights aren't on the same circuit as the headlight(s). To get the car legal for
night driving, replace the broken headlight (both need to work to drive legally at night),
and tackle the running lights as a separate project.
First thing to check on the running lights - the bulbs. Why: incandescent light bulbs that are
on at the time of an impact cannot withstand the impact and the filaments can fail, so fixing
your rear running lights may simply be a matter of replacing the two light bulbs.
If the bulbs are ok, your accident has caused a wiring fault, which you will locate
between the ignition switch and the rear running lights.
Here's the manual on how to replace the headlight:1993 Buick Century
HEADLIGHT REMOVAL & INSTALLATIONNon-European Style
See Figures 1, 2, 3, 4 and 5
Remove the headlamp trim panel attaching screws.
Fig. 1: Remove the headlamp trim panel attaching screws
Remove the headlamp bulb retaining screws. Do not touch the two
headlamp aiming screws, (at the top and side of the retaining ring), or
the headlamp aim will have to be readjusted.
Fig. Fig. 2: Remove the headlamp trim panel assembly from it's mounting location
Using a screwdriver, remove the retaining ring screws.
Fig. 3: Using a screwdriver, remove the retaining ring screws
Fig. 4: Once the retaining ring screws are removed, the ring can be pulled away
Pull the bulb and ring forward then separate them. Unplug the electrical connector from the rear of the bulb.
Fig. Fig. 5: Pull the bulb out and unplug the electrical connector
Plug the new bulb into the electrical connector.
Install the bulb into the retaining ring and install the ring and bulb.
Install the trim panel.
Replacing the European style headlights is a slightly different procedure.