I can tell you why it’s happening but it’s not going to fix your problem. If you use the share sheet it basically sends a copy of the file. This is true for any file type - it’s not erasing it from the original location, it’s creating and importing a copy of the file to the location you asked. So DTTG is reporting the correct creation date, which is the date and type you created the copy. The original file remains in its original location (in this case the Files app).
This seems a reasonable mechanic, I suspect a lot of customers would be annoyed if files got removed from the original location completely when you used the share sheet.
I can’t see a way round this. I tested an alternative workflow for you using the “Move” function in the Files app to move the file from iCloud to DTTG, but that’s explicitly a copy mechanic as well (Apple name it as such and I’ve confirmed this). As far as I can tell the only way you’d get what you want is by moving the files on a Mac, which wouldn’t have this copy mechanic.