Why automatic gear change hanging on - 2005 Audi A3

Yes, it is a known issue

https://www.google.com/search?q=vw+jetta+2.0+tfsi+rev+hanging

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no car stated (Gaia what? er Toyota )
2.0Liter told later
no transmission type, AUTO IT IS. NOT STICKSHIFT.

I guess dragging means delayed shift in the automatic>?

why not ask your autoshop this,.>?>?> as 99% do.

if any automatic fails to up shift the speed is low
speed low cause the speedometer to read low.
not upshifting can be cause by 2 things, weak engine or bad transmission.
you will never learn or discover why only your shop will. BUT I did answer below. and can expand each line by 100 fold if need be.
and more if the year car was told or what transmission model is there, some cars had 2 choice or more back then.

CHECK YOUR TRANSMISSION FLUID WHILE IT'S RUNNING IN PARK,COULD BE THE PROBLEM,IT'S IT'S REAL LOW.

check transmission fluid level, it could be low on fluid causing it to hang up and not shift properly.

stick? broken syncronizer.. automatic.. valves hanging up inside the valve body..may nned fluid and filter change..

torth converter

sounds like u have two get reverse gear in the gear box tensioned up or repaired

Have the trans filter& gasket changed. Valve body maby sticking fresh fluid and filter might help.

automatics can be changed to manual but its not that easy. Not only do you have to change the gear box you have to change all related parts (ie: driveshaft, flywheel, ujoints etc.) you then have to hang a clutch petal mount a slave cylinder. Might want to rethink this project.

I would take it to a shop for service. Change fluid and filter. Might want to add an additive to it but clean fluid should clean it up. My experience is just clean fluid will work most of the time.