On the Original Contenders with the rotorary selector switch on top of the hammer the triggers are fairly easy to adjust.
Yes the Screw in the front portion of the trigger adjust the amount of engagement. Turning this screw IN decreases the amount of engagement. As per the T/C instruction manual the way to check for proper engagement is as follows:
With the Chamber EMPTY, open the Barrel, then Cock the Hammer. With a sharp snap of the wrist snap the barrel shut (ie holding the Contender by the pistol grip, snap your wrist upwards sharply causing the barrel to snap shut). If the hammer falls when this is done the engagement is too fine, and must be increased.
The screw through the rear of the Trigger Guard is for limiting the overtravel of the trigger after the gun fires. Set too fine will cause the trigger guard to move with the trigger pull and will actually increase the pull weight (and could cause an unsafe condition).
To adjust the pull weight there are a couple of thing to do, clip a coil off of the trigger return spring, or replace the trigger return spring all together (the spring from a long tire valve stem is about the right diameter, but lighter in weight).
To remove the trigger assembly from the gun, first take the barrel off of the frame. Next remove the grip. Once the grip is off the screw that goes up through the rear notch in the trigger guard needs to be removed (there is a spring around the outside of this screw that goes between the reat tang of the trigger guard and the frame). Once this screw is removed, carefuly punch out the retaining pin that goes through the sides of the frame where the trigger pivots.
Once this pin is removed the trigger assembly can be dropped out the bottom of the frame. Look at things carefully. The "Striker" on the rear engages the engagement notch on the trigger, and when the trigger is pulled the striker is released and in its travel trips the hammer to fire the gun.
With the trigger housing out of the frame you can actually test the trigger pull weight buy setting the striker in the trigger engagement notch and pulling the trigger. This will be the trigger pull weight just like if everything was assembled in the frame.
If a lighter pull weight is desired, carefully push out the trigger pivot pin. Once the pin is removed "VERY CAREFULLY" slide the trigger to the rear of the trigger guard making sure not to let the Trigger Return Spring and Cap fly free. With a little wiggling, and tilting the bottom of the trigger forward the trigger can be freed from the trigger guard.
Now if you are satisfied with the engagement and crispness of the trigger pull you just need to work on the spring. However if creep is still detectable a little polishing should fix things up. Carefully, very carefuly polish (just polish) the engagement surface of the trigger notch, and the engagement surface of the striker (I generally punch out the roll pin holding the striker in place to polish it, and polish the sides of the striker where it pivots in the process).
That is about the extent of it. If you are familiar with Triggers and how they operate it should be fairly easy to do. However if you are not familiar with Triggers and how they operate it should be a simple matter for your gunsmith to get you a great trigger on your Contender.
The Contender (not the G-2) had in my opinion one of the BEST triggers of any gun period. The design is such that there is very little travel to the trigger to fire the gun, and they can be made SUPER CRISP and quite safe with trigger pull weights of down to a pound and a half.
Good luck.
Larry