While driving depressing accelerator to the floor car bucks down shift to upshift won't stay in lower gear. Normal acceleration shifts fine

this is normal for the trans to downshift as the engine and trans are working together to handle the acceleration of your car and as your car's speed increases the trans will adjust again (EG: shift gears) this is more of a weight to acceleration sort of thing or rather weight to power sort of thing

'Passing gear' is a term that means the transmission is down-shifted to a lower gear (e.g., from 5th gear to 4th gear) for improved acceleration to pass another vehicle. There is no physical component called a passing gear in the transmission.
To engage the 'passing gear' with an automatic transmission should require nothing more than pressing the accelerator pedal down firmly. The transmission should downshift automatically to the next lower gear (sometimes it will downshift two gear ratios, depending on your speed and how far you depress the accelerator pedal).

Sounds like you may have fuel filter problem. you may want to check the pump and filters located in or near the fuel tank.
Go directly to the source ( Fuel Tank, Pump, Filters, Fuel Pressure Regulator and Injectors )
You may have water or grit in the fuel
Hope this helps

transmission rebuild time the clutch disc are burnt all ready

1. check transmission fluid level or if you can change the fluid & filter.
2. check throttle position sensor (tps).
3. if fluid level & tps are ok & still not upshifting automatically, temporarily (so you can take it to a transmission specialist) , try to manually shift. starting from dead stop, shift to D1. at about 25mph shift to D2. (be gentle on the accelerator & don't over rev on each gear. release accelerator pedal & don't step on the brakes while shifting from D1 & D2).

Worn clutches or solenoid issues if it is an Alison style box.

Hi Seth, Check out your engine and gear box mountings! Regards John

On most of these type vehicles, with the heavy duty and lower geared transmissions this is normal, when accelerateing hard it will stay in lower gear longer...

E for economy, and S for sport. The button allows you to switch from normal mode E economy, to S sport mode for better acceleration and quicker down-shifts when driving in a more agressive or performance manner. The S mode will make the transmission shift at higher revs before engaging the next gear, in order to give better response. Likewise when you enter a corner and let off the throttle (gas) , most transmissions will often change to a higher gear. This is not ideal for spirited driving, because you want the revs up a bit and to stay in the lower gear to accelerate out of the corner after you have cleared the apex. For everyday driving the sport mode keeps the car from up-shifting and down-shifting too much in traffic or on windy roads etc. The emphasis is on performance rather than ultimate fuel economy in Sport. Try it out next time you feel sporty and fell the difference. But for best gas mileage leave the car in E mode for economy.

from the car manual:

Tum the overdrive switch on for good fuel
economy and smooth driving. If engine braking
is needed in the "D" range or if repeated
upshifting and downshifting between the 3rd
and 4th gear is needed when climbing a
gentle slope, it is recommended that the
overdrive switch be tumed off. Tum the
overdrive switch back on immediately afterward.

When the overdrive switch is tumed on, the
transaxle will automatically upshift to the second,
third and overdrive gears. When the overdrive
switch is tumed off, the transaxle will not
upshift to the overdrive gear. For normal driving,
the selector lever should be left in the "D"
position and the overdrive switch tumed on.
If you need to accelerate rapidly, press the
accelerator pedal all the way to the floor. The
transaxle will automatically shift to a lower gear,
depending on the vehicle speed and load.