Make it clear, copying files from your Mac to a USB drive should not delete the original files. However, the files are unexpectedly disappearing from your Mac, to troubleshoot it, try iBoysoft Data Recovery for Mac, professional data recovery software to recover lost files on Mac.
When you copy folders from your iMac to any external storage media such as USB flash drives, SSDs, HDDs, SD cards, etc., the copied files will display on the external hard drive and the original files remain on your iMac. Perhaps, you mistakenly used the Cut command instead of the Copy command.
The copy command makes a duplicate of the specified data, whereas the cut command removes the selected data from its original spot. When you copy and paste on Mac, it doesn't affect the original files, while if you cut and paste on Mac, the selected files are moved from the iMac to the USB drive.
As you said, you haven't deleted them and emptied them from Mac Trash, and there is no Time Machine backup available to use as the Time Machine is stuck on preparing backup, then, you can give iBoysoft Data Recovery for Mac a shot. It features a high recovery rate, a risk-free environment, and an easy-to-use UI.
You need to boot your Mac into recovery mode to recover data from the Mac hard drive, keep reading on How to Run iBoysoft Data Recovery in macOS Recovery Mode.
