Front left running light burns too bright and left rear blinker blinks faster.

SOURCE: Front L/R turn signal lights remain on.

bad ground or check the bulbs mack sure none are blown.

SOURCE: 1993 Heritage Softail Classic motorcycle. Front

First, I'd try replacing the bulbs in both the front and rear turn signals. See what happens, that's pretty inexpensive to try.

The turn signals are controlled by the turn signal cancelling unit. You didn't mention if the problem was on just one side or if it was on both sides of the bike. I'd try to find someone that had a known good unit that would allow me to try it. The reason, they're expensive. The last one I installed cost nearly $150.

SOURCE: i have 1993 harley softail

Hi, technically i'd have you to to first of all trace the circuit to learn what's causing the problem before contemplating on a new module and using a multi meter to run the test as intended by you is a good idea. That will determine the condition of the turn signal switch and contacts.

A tip about how the Turn signal works....

The way the turn signals on your bike work is what is known as "Cheap and Nasty".
All the manufacturers do it this way. The resistance in the turn signal circuit is acutally very important to how they work. When you set the indicator switch to turn, you make a contact in the handlebar switch, and this provides either power, or a ground to the indicator relay. This is usually a three terminal device located conveniently where you can't get at it. The relay picks up, and this provides power through the relay coil to both front and rear indicator lamps. (Turn signals for the US residents.) The current drawn by the bulbs in the indicators heats up a bi-metal strip in the relay, and when it gets hot enough, it bends the moving relay contact that is providing power to the bulbs away from the fixed contact and the bulbs go out.
This bi-metal strip then cools down. It contracts and the moving contact makes the circuit with the fixed contact and the bulbs come on again. And so on. So the lights flash. And will continue to do so until you remove the power or gound from the input to the relay.

SOURCE: Right blinker wont blink. Light is on but wont

turn signal module or (TSM) is bad assuming you are using OEM turn signals.

SOURCE: I have a 1994 Fatboy that the right turn signals

Probably a bad Turn Signal Module located under your dash speedo...remove your dash and speedo then you will access to the module.....