I have to let up on the accelerator to get my transmission to shift into second gear

Sometimes you can only diagnose a faultwhen it is their / happening ....try earth straps around engine remove , clean and refit

Sounds like a vacuum leak probably at the modulator valve on or close to the transmission.

This may apply. ...............................

Carry out the adaptation drive cycle road test:

Note: This process must be carried out in normal mode (not sports mode). On a flat road, the transmission fluid temperature must be above 60° C (140° f) and below 100° C (212° f) (vehicle at normal operating temperature will ensure that the transmission fluid temperature is above 60° C (140° f)).
9. Accelerate from rest with light throttle (do not exceed the upper rev limit in appendix 1). 1-2, 2-3, 3-4 shifts must occur with engine speed between the following limits:.
Note: Do not move the accelerator pedal during shifts.

• 3.0L VIN ALL 1800 - 2000 rpm
• 4.2L N/A Prior VIN M96321 1450 - 1600 rpm
• 4.2L N/A From VIN M963221550 - 1700 rpm

10. Continue to accelerate gently to 50 mph (80 km/h) so that transmission shifts into 5th gear.
11. Go into overrun and let the transmission shift down into 4th gear without braking.
12. Brake gently to a standstill and hold footbrake on for at least 15 seconds.
13. This process should be carried out at least five times.

Page 2 of 3 Bulletin Number S307-17
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Most likely your shift cable (from the selector to the transmission) is worn, maybe needs lubrication. Mine had a similiar problem, but had to shift it into neutral and back up to get it in reverse. A new shift cable solved the problem.

i had that problem on a 93 escort and it was because the woodruff key that holds the harmonic balancer in place was broken, the poopy acceleration was because the crank sensor gets its "signal" from cogs on the harmonic balancer, which did not turn at the same speed as the crank shaft because of the drag on the crank pully (crank pully and harmonic balancer are same part)

You rear planetary set is worn from a restricted cooler.

Try this relearn procedure hopefully it will help

Vehicles equipped with engine or transmission computers may require a relearn procedure after the vehicle battery is disconnected. Many vehicle computers memorize and store vehicle operation patterns for optimum driveability and performance. When the vehicle battery is disconnected, this memory is lost. The computer will use default data until new data from each key start is stored. As the computer memorizes vehicle operation for each new key start, driveability is restored. Vehicle computers may memorize vehicles operation patterns for 40 or more key starts. Customers often complain of driveability problems during the relearn stage because the vehicle acts differently then before being serviced. Depending on the type of vehicle and how it is equipped, the following complaints (driveability problems) may exist:

Harsh Or Poor Shift Quality
Rough Or Unstable Idle
Hesitation Or Stumble
Rich Or Lean Running
Poor Fuel Mileage

These symptoms and complaints should disappear after a number of drive cycles have been memorized. To reduce the possibility of complaints, after any service which requires battery power to be disconnected, vehicle should be road tested.

Vehicle Preparation

Ensure all components are connected. Ensure transmission fluid level is correct. Warm engine to normal operating temperature. If vehicle has been repaired, perform KOEO and Continuous Memory Code Self-Test and ensure fault codes are not present.

Computer Relearn Procedures

Idle Relearn Procedure

Place automatic transmission in Park or manual transmission in Neutral position. Ensure the emergency brake has been set and all accessories are turned off. Start engine and bring to normal operating temperature. Allow manual transmission vehicle to idle in Neutral for one minute. Allow automatic transmission to idle in Neutral for one minute and then to idle in Drive for one minute. Once initial relearn is complete, process will be completed during normal driving. Always complete the procedure before returning the vehicle to the customer.

Drive Cycle Test (AOD-E, AXOD & AXOD-E TRANSAXLES)

With transmission gear selector in Overdrive, moderately accelerate vehicle to 50 MPH for a minimum of 15 seconds. Transmission should be in 4th gear. While holding speed steady, lightly apply and release brake for about 5 seconds. Stop and park vehicle for a minimum of 20 seconds with transmission gear selector in Drive. Repeat procedure 5 times. Always complete the procedure before returning the vehicle to the customer.

Drive Cycle Test (E4OD)

With transmission gear selector in Drive, press Overdrive Cancel Switch (LED should light). Moderately accelerate vehicle to 40 MPH for a minimum of 15 seconds (30 seconds above 4000-ft. elevation) Transmission should be in 3rd gear. While holding speed steady, press Overdrive Cancel Switch (LED should go off) and accelerate from 40 MPH to 50 MPH. Transmission should shift from 3rd gear to 4th gear. Hold speed steady for 15 seconds. While holding speed steady, lightly apply and release brakes enough to turn brake lights on. Maintain 50 MPH for about 5 seconds. Stop vehicle for a minimum of 20 seconds with transmission gear selector in Drive. Repeat procedure 5 times. Always complete the procedure before returning the vehicle to the customer.

transmission fluid. you need some. ours did the same. fix it fast, before you damage the transmission.

SHift solenoid pressure switch

first thing you need to do rule out vacuum to shifting module make sure there is no leaks you usually can detect by a hissing sound or high pitch whistle check your shifting module on your transmission providing transmission screen and filter has been maintained as for the second part gm powersteering pumps have a tendency to whir if a new belt has been installed until it breaks in