For what it’s worth, I’ve also observed a bug where using the sharesheet with a PDF in Safari “dims” and freezes, without saving the pdf to DTTG. I’m on iPad 16.3.1 and DTTG is 3.6.3. The bug is unique to DTTG and is initiated by selecting DTTG in the share menu. Essentially nothing happens once you click on it. In my case, I’ve opened DTTG to see if the file magically made it there anyway (it doesn’t) and on returning to Safari I’ve been able to click the sharesheet again and it’s worked the second time.
I’ve not had time to investigate it properly (or report it) but thought I should leave a comment to confirm there is a bug on the current version.