How do you replace the rear running light bulbs?

Tony, our TB's only have 3 bulbs, 1 brake, 1 running, 1 backup.
If you happened to be playing with NEW LED bulbs (from local retail location) for the brakes, guaranteed you blew the fuse under the rear seat. It takes a special CK socket for the brakes that no one carry's. If you're only dealing with normal incondecent bulbs, I'd still check the fuse under the rear passenger seat.

Mr Pinske,

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 & INSTALLATION
Non-European Style
See Figures 1, 2, 3, 4 and 5

1. Remove the headlamp trim panel attaching screws.

Fig. 1: Remove the headlamp trim panel attaching screws

1. 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

1. 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

1. 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
To install:

1. Plug the new bulb into the electrical connector.
2. Install the bulb into the retaining ring and install the ring and bulb.
3. Install the trim panel.

Replacing the European style headlights is a slightly different procedure.

The outer lights are dual filament bulbs, meaning they work 2 separate functions, brake and running lights. Either filament can burn out separate from the other, thus creating the situation you have. Replace the bulbs and your problem should be fixed.

2 fuses, for park lamps, 6C 25 amp and 8C 10 amp

you replaced the bulb with the wrong one i.e. a two filament bulb with a one filament bulb

Inside the boot is a plastic cap embeded in the boot liner. Remove the cap to reveal a screw. Undo the screw and the whole rear light cluster will come away in a sideways and not rearward direction to reveal all of the bulbs. Hope this helps.

possibility of a bad ground or a corroded wire

If the driver's side tail light is working, it's not a fuse. Use a test light to check for power at the bulb holder. One terminal will be for tail light, the other for brake/turn light. If the is constant power for tail light, replace bulb. If no power, trace wiring from bulb holder to driver's side junction. Probable broken wire between the two tail lights.

Check the 10 amp "Tail" fuse. This controls the running lamps.
Note that the rear bulbs are "dual filament" bulbs, and even though the brake lamp feature works, the running light feature in the same bulb can be bad.