No blinkers unless 4 way on - 1998 Lincoln Town Car

Turn the 4 way switch on and off several times. Sometimes the connection that goes from turn signals and emergency
lights get dirty.

Check the fuse then check the ground... sounds like a broken wire or bad ground... most likely wire... unless blinkers are not working in the front also then that would most likely be the blinker switch

Sounds like the ignition switch. It is a very common problem for those vehicles.

This sounds like the ignition switch needs to be replaced. These are very typical symptoms of a common problem in these vehicles. It is probably not related to anything you did, but be sure you don't have a lot of weight hanging from your keyring.

It's probably the ignition key switch located on the left side of the steering wheel. This is a fairly inexpensive part, but pay the 2 or 3 bucks more for a name brand switch as the really cheap ones only last a short time.

Replace the BCM.

You said left and right, not that it really matters unless both are dead.
Have you checked the socket to see if you have power and ground on that circuit ?

The do not run off the same fuse but they do run off the same blinker. Try turning on the key, click the left/right (check both) turn signal and then operate the 4 ways on and off at the same time. Work the 4 way switch back and forth 10 to 25 times and see if you get any short winks or blinks from the system.
Turn signals are usually on their own fuse in the glove box and hazard are on with the brake fuse. If this does not work, I would look at the blinker. It is possibly bad. Check each pin of the blinker and see if there is power there when you turn on the 4 ways or turn signals. Replace Blinker. Make calls to parts stores and dealer for location of this blinker.

Defintely sounds like the turn signal switch is bad.
If the flasher was bad, your right blinker wouldn't work.

Your gonna have to remove the blinker panels to replace the lights. I'm assuming your talking about the clear rectangular blinkers on the front? If so, start with the orange blinkers on the sides, 2 screws, under that blinker will be 2 more screws to remove the mount that holds your blinkers in place. Look around the 4 corners of your headlight, you'll find more screws. Once you've unscrewed everything you'll be able to remove the blinker mount thing (square-ish piece of plastic) and then you should have access to the clear rectangular blinkers on the front. Very easy.