Traction Control comes on while shifting into Drive on dry surface.

Most 4WD systems are designed to be shifted in and out "on the fly". It is less stressful for the drive train doing it while in park however. 4WD does not like cornering - especially on "grippy" surfaces such as dry asphalt (why would you need 4WD on dry asphalt, anyway?). Dry asphalt is really tough on 4WD as the the tires may have minute differences in circumference as a result of wear and since all are spinning at the exact speed (even going straight ahead), those difference add up and strain the drive train. Slippery surfaces allow the tires to slip away those differences as soon as they begin to appear. Even wet pavement allows this to happen, tho not as easily as grass, dirt, snow or ice.

Drive in a tight, complete circle on dirt and you'll see that there's a shorter path of tracks for the inside tires than those on the outside. Yet, the 4WD locks all wheels to turn the same amount. The jerking during the cornering that you're feeling is the inside tires trying to spin to "catch up" to turning at the same rate as the outer tires. See the video linked below.

Check your manual and you'll likely be cautioned against running 4WD on dry pavement for these reasons.

If it comes on occasiionally, you obviously are driving on some slick surface. When it senses the slip, it automatically engages the traction control.

Even minor slippage will engage the traction control, like turning a corner, and one wheel slipps a little. If the light is constantly on, you have hit the switch for the Traction Control.

The traction control and transmission shifting would involve the body computer more so than the engine computer.
Have you had the body computer scanned for trouble codes ?

if you are driving in 4H on hard surfaces and just flip the switch to 2H still on the hard surfaces it will react exactly as you describe
when in 4H as there is no internal diff action in the transfer case , a condition known as torque wind up occurs and places heavy loads on the drive line components thereby locking the gear teeth and that will not allow the separation on selecting 2W drive
if at some point one wheel looses traction for a split section, that torque wind up is reduced and the pressure on the actuator slams the selector out of engagement
4H should only be used in slippery conditions , mud, dirt ,snow,ice, grass etc where the wheels loose traction and allow that wind up to dissipate
if you continue to drive on hard surfaces in 4 H then transmission failure will occur earlier than can be expected

That is your Antilock Brake/Traction Control computer going into what I call "False antilock mode" This is almost always caused by abad wheel speed sensor. Your vehicle is equipped with 4 of them. To diagnose this problem, you must drive the vehicle with a scanner hooked up that can read the wheel sensor data and find out which wheel sensor is not working properly. Sometimes (but not always) you get lucky and the Antilock Brake Control Module will set a code to tell you which one.

TRACTION CONTROL: An enhancement of an existing ABS system that prevents wheel spin while accelerating on wet or slick surfaces. It uses the same wheel speed sensors to monitor wheel speed during acceleration, but requires some additional control solenoids and a pump to apply braking pressure to control wheel spin. The traction control system brakes the drive wheel that's starting to spin to shift torque to the opposite drive wheel that still has traction. Most traction control systems only operate at speeds up to about 30 mph. Additional control strategies that some traction control systems use to limit wheel spin include reducing the throttle opening, upshifting the transmission, retarding spark timing and deactivating fuel injectors.

Hope helps.

I would recommend you have the vehicle scanned for trouble codes. You can get this done at no charge at your local Auto Zone or Advance Auto Parts store.
Record the code numbers and reply back to this message and we can go from there.
Based on your description, I would guess you may have a HWSS ( hand wheel speed sensor) issue or possibly an ABS wheel speed sensor problem, again, this is only a guess until you have the codes read.
Let me know.
Regards,

Traction control helps limit tire slip in acceleration on slippery surfaces. In the past you had to feather the gas pedal to prevent the drive wheels from spinning wildly on slippery pavement. Many of today's vehicles employ electronic controls to limit power delivery for the driver, eliminating wheel slip and helping the driver accelerate under control. Don't turn it off.

that is what it is desgined to do - reduce acceleration and apply the brakes intermittently when the wheels spin. You have a switch you cn push to disable it. You need to disable each time you start car. See owner manual.

06-10: Traction Control 1996 Contour, Mystique Workshop Manual
DESCRIPTION AND OPERATION Traction Control System The optional traction control system is designed to control wheel spin when accelerating on slippery or loose surfaces. Traction control system is available only with anti-lock brakes. The traction control system includes the following components:

• Hydraulic control unit.
• Throttle control actuator (9N825).
• Traction control switch.
• Traction control warning indicator.
• Front brake anti-lock sensor (2C204).
• Rear brake anti-lock sensor (2C190).

The traction control system operates as follows:

• System functions by transferring torque from the spinning wheel to the wheel with the most traction.
• This is achieved by applying the brake at the spinning wheel through the hydraulic control unit or reducing the throttle opening.
• Traction control braking and throttle control occur at vehicle speeds up to 50 km/h (30 mph).
• At vehicle speeds above 50 km/h (30 mph), only throttle limiting control is used.
• If speed control is operating when traction control system engages the traction control system will disengage speed control through the use of a speed control relay. This action is the same as applying the brake pedal with speed control engaged.

1996 contour
Traction Control Switch The traction control switch is located on the instrument panel, left of the steering column.