This issue is due to problems in the underlying packages and has been reported:
rstudio/DT#1123
https://datatables.net/forums/discussion/78109/bug-is-it-possible-to-fix-incremental-cell-dragging-autofill-extension#latest
As a temporary solution, you can use an earlier version of DT:
remotes::install_version('DT', '0.30')