Your description is ........ Confusing. Are you saying the guns fired from half-cock??? Or, if you lightly touch it, it just drops the hammer into the half-cock notch??
Anyway, the Single Action is not designed to have A really light trigger pull. I would guess, if you pulled the gun apart, you would find the trigger sear is A. Badly worn. B. Rounded over at manufacture. Either way, the trigger sear would need to be recut and hardened.
You may also find the full cock ledge on the hammer is worn. If so, the ledge would need to be corrected and re-hardened.
It may be a combination of both.
You should also do the "pull hammer forward real hard" test. You should NOT be able to pull the hammer forward off the trigger sear.
The Single Action was actually designed to function correctly with a 4 - 5 lb trigger pull. The trigger can be adjusted to function correctly with a lighter pull, but, and a BIG but, the angle of the dangle relationship of hammer ledge and trigger sear must be correct.
It's not a kitchen table job and is best done with a special jig.
Coffinmaker