Shifting on the X-axis while reprojecting Gauss-Krüger zone 3 to UTM zone 33N

I’m having trouble converting geodata from Gauss-Krüger Zone 3 to UTM 33N.
See attached image.

I don’t know why, but there seems to be a shift on the X-axis. I found a similar question here but am not sure how to do this in QGIS:
https://sourceforge.net/p/saga-gis/discussion/790705/thread/a824273a/?limit=25#567a

The left map is from here: https://gadm.org/download_country_v3.html

The right map is from here: https://opendata.dwd.de/climate_environment/CDC/grids_germany/multi_annual/precipitation/