At idle Engine accelerates when moving the steering wheel

You don't need to worry about the steering wheel with this as they are not connected to each other in any way.
Are you getting any changes at all? OK, with the engine running, does the car feel the same when you select D or R? Do you feel the car change slightly as the gear engages and the sound of the engine slow slightly as the engine connects to the transmission?
Or does the car feel the same and the engine sound doesn't change. Is it only the Shifter stick moving and the indicator light changing?

If you feel the truck lurch forward in drive and backwards in reverse, I would suspect stuck brakes or the parking brake is stuck on due to a bad cable or something. If it doesn't lurch, try giving it some gas and see if it feels like it wants to go back or forward. If no movement at all, i would check the transmission fluid (Remember that you check this while the engine is idling). If the fluid level is in the cold range, i would suspect a failed transmission pump or a failed torque converter. If this is 4x4- check to make sure that the transfer case is in 2hi and not in neutral.

Engine running ask someone to turn steering wheel for you . Look for power steering hose move like there is pressure. and noise from power steering pump . if not change power steering pump
You don't really need power assistance when moving on the road ... where you need it more specially in parking or not really moving
hope it helps

It's not a malfunction- this particular auto's idle rpm speed is low and it simply lacks the power to hold on the gradient of the slope- until of course you step on the accelerator pedal.

There is a very brief lag time from the moment your foot comes off the brake pedal to the time it reaches the accelerator pedal; such that by the time the torque converter is able to "perceive and respond' to your brake pedal to accelerator pedal movements, the car will- by the vary nature of gravitational force move in the direction it is being pulled until a force of equal or greater resistance is applied to either hold it in position or move it in the opposite direction.

By its very definition, a torque converter is a type of fluid coupling that is used to transfer rotating power (the engine) to a rotating driven load (the drive train)- that is the transmission, drive shaft, axle gears, and wheels that the tires are mounted on. [The transverse engine set-up is different than an in-line engine design, but the operating principles are basically the same]..

Automatic transmissions work on the same principle of a water wheel. As the current force of the water drops into the vanes of the wheel, it is converting that water force or pressure into useful forms of power as the wheel spins.

If the current flow of the water is so slow that the paddles of the water wheel are unable to detect any force, the wheel does not move- therefore no power-torque output.

Likewise, if the engine is idling too slow for the vanes in the torque converter to detect any push the transmission into power-torque output, then the car is going to freely go into the gravitational direction of the slope.

id go for chicken feet and blood and voodoo , sessions.....loL

first off, i see an illegal fast idle ,
and all A/T trannys do that, when the engine rpm fraks up.
to move car, due to the stall speed rules of the clutch torque converter (aka ,the fluid clutch) wiki clutch stall speed.

that A/T weakness, is why folks run over kids. (the lame excuse)
starting a car with foot off the brake is , folly, thus the lock all outs called park lock features.

i can not follow your sequence of events.so will ask questions.
1: I start the engine, its don't want to but does. (ever tune it up?)

2: Does the CHECK engine lamp go out (CEL/SES)? running.
for sure, if the engines stalls many lights will glow in the cluster but that is normal, see? for any stall.!
The Charge light comes on anytime the Alternator goes dead
or hits near 0 RPM (stalled)
so are you saying at 3000 rpm, the charge lamp came on.
/??????
battery lamp is not battery, its really a Charge lamp.
the op guide states that too.
on my car I have a battery gauge too, and shows VOLTS.
at idle its 14v. its 14v at all rpm. if less, say so.
I have 2004. GV. with full manuals. even factory FSM. ++++++

ok , in drive, no acceleration, are you saying ,
1: im in A/T drive.
2: the engine is idling at 800 RPM hot or near 1500 cold
3000 is wrong, dead wrong, you in ALASKA, state location(gen)

i increase the RPM with my right foot and the car don't move
in DRIVE
as RPM rises above 1500 RPM, you do have a tacho , i see that.
if it does not move with hgher RPMs, the trans is bad or its TCM is bad.
on your car the TCM is hidden deep inside the PCM. back in 1997.

the PCM runs the engine and tranny today.
if something lies to the TCM/PCM the tranny will not go into drive.
on newer 96+ cars, you see the check engine lamp glow.
running (not stalled, stalling this CEL glows all the time)
but running, the CEL glowing means Faults found, scan me,. please.
eg:
so you scan it, and find DTC's stored (many?) eg.... in english...
the transmission selector switch is bad. or worse.
not using a scan tool ,means ($7 cost) means you miss all the facts. hard cold real facts.
the ECU can not shift any car with a bad selector switch.
so throws the DTC error.
it can be even be a loose selector cable, cave man simple.

recap.
does the CEL lamp stay out running at all times.
no, ? then scan it.
do not post car problems with stored, or pending DTCs on any car newer than 1995. or its just wild guessing.. mostly.

often times the engine needs to warm up to normal operating temperature. this varies from makes and models. if you cant wait that long, having it idle for approximately 5 - 10 minutes should at least lessen the acceleration problem, but the longer you wait the better it will be on the engine.

recheck balance and inspect threads can have a possible bad thread belt on tire

on most power steering pumps there is a single wire idle up switch to compensate for load caused by pump switch is on pump check connection also try to find a clear parking lot drive in slow circle steer all the way see if engine dies while moving at slow speed will use less effort if engine doesnt die then check the p/s idle up switch

would need to have a play but i would go for the idle control valve first ,then maybe the power steering where their is a sensor to increase idle when wheel is turned whilst stationary ,the sensor is on the rack housing where the steering wheel enters the rack

try looking for an airleak in the hose from the inlet manifold to the brake reservoir,