1991 Park avenue, Tail lights and license's plate light's stopped working turn signal's hazards and brake lights still functional. but no tail lights or license plate lights

LED's have resistance than incandescent bulbs, that causes a problem, so best thing to do is get ones that come with the correct resistors.

No such thing , tail lights get power from headlamp switch . When you turn on park or head lamps . tail lights light up . License plate lights come on as well as side marker lamps . Unless you thing of the turn signals an hazard lights . There is a flasher relay for that .
Turn/Hazard Flasher
At the left lower instrument panel, near the steering column

It's marked LP PK R 15 amp , it's located in the instrument panel fuse box . IP Fuse Block
Under the right side of the IP, near the rear of the IP compartment

LP PK R Fuse 15
15 A
Remote Function Actuator (RFA) Module, License Lamps, Tail/Stop Lamps, Tail/Signal Lamps, RH Side Marker Lamps, RH Front Park/Turn Signal Lamp

I'm checking into that same problem for a early 90's buick ( it's a GM also ) , I've checked the obvious ( bulbs, flasher,fuses,etc) all where fine. I assume your front turn signals work if you manually operate the turn signal switch or put on the hazard switch? The Buick's rear lights won't come on even with the hazard switch on. When you say one tail light has been out for some time, do you mean the brake/signal light or park/running lights? The Buick was like that with one side out for a while ( brake/signal light) but both the rear park/running lights work. Now, I've narrowed the problem down to the turn signal switch. I'd bet money on it. I just have to dig out my steering wheel puller, bite the bullet and get 'er done. You should be carefull if you're attempting that repair yourself ( AIRBAG ), use a manual. ( You could do a quick test by trying the turn signal while you pull the bottom of the arm toward you,move some of the wiring in the column near the arm,etc. I did and the signals started working. Could be the switch,dirt or a broken/loose wire. there's a lot of wiring going through the column and a lot of movement ( turn signal,high beam,cruise, then add tilt) Hope this helped.

I do not know for sure, but it sounds like you have a short some where. You might want to start looking for bare wires around that tail light that first started giving you all of the problems. I hope this helps you solve the problem

Hope this helps your out. I know the diagram and locator is a bit blurry but hopefully it gets what you need.
Drivers kick panel:
Front wiper 20A - Front wiper system Cigar 15A- Cigarette lighter Audio 10A- Audio, remote control mirror Air Bag 10A -Air bag diagnostics Stop lights- 15A -Rear brake light, high mounted stop light ECU B+ 10A - Shift lock actuator, transaxle controle module, radio, engine control module Room 10A - Interior light, clock, ignition key reminder, rear cargo area light switch ABS 10A- Anti lock brake system control unit ECU IG1 10A - Transaxle control module, shift lock actuator, ABS control unit, DRL unit Meter 10A - Instrument panel, back-up switch, electronics time control module, inhibit switch Turn signal 10A- Hazard switch ,front and rear turn lights, turn indicator IGN1 20A - Ignition switch ABS 30A- ABS pump motor TNS 30A- Tail light, license plate light COOL/FAN 30A- Cooling fan Cond/Fan 20A- A/C condenser fan Starter 15A- Start signal , Fuel pump Blower 25A- Blower Motor SR/ACC 10A - Ignition switch Hazard 15A - Hazard and turn signal light D/Lock 25A- Door locks ABS 30A - Anti lock brake system control P/W RH 20A- Power window motor right hand P/W LH 20A- Power window motor left hand IGN2 25A- Ignition Switch
Engine compartment :
Head 25A- headlights IG 2 30A- Ignition switch IG coil 15A -Ignition coil, condenser,generator IC regulator, diagnostic connector. O2 down 10A- Rear O2 sensor O2 up 10A- Front O2 sensor Fuel pump 10A -Fuel pump unit Injector 10A -Fuel injector ,camshaft positioning sensor, purge solenoid valve, EVAP, IAC actuator, Mass Airflow sensor, EGR stepper motor. fuel pump relay coil, vehicle speed sensor. A/Con 10A - A/C compressor controls, cooling fan BTN 30A- Brake light , high mounted stop light. Tail light LH 10A- License plate light, tail light , rear combo light, instrument panel illumination Tail light RH 10A-License plate light, tail light , rear combo light, instrument panel illumination Head low 15A- Headlight Head Hi 15A-Headlight Horn 15A- Horn

Inside kick panel

With light switch on, test the wires leading up to the mentioned lights with a test light(cost about 5 or 6 bucks at wal-mart) that way u can eliminate any corrosion questions and get to the source quicker, if test light indicates that the wire is hot, than check connection areas for any wear and tear and possible carbon corrosion. Check lightbulbs, my license plate light stopped working and wire was hot, but wouldnt turn on due to the housing base, got too dryrotted and wouldnt function propery, had to rig it up, but works now.

check for faulty signal light flasher located under L/S dash. Separate flasher for hazard lights and signal lights.