That's a concern even for files which do have an extension; I have set & would recommend setting the transfer to "Binary" for all, rather than "auto" of any kind. Transferring "text files" which are actually .php, .htaccess, etc., via binary will still deliver them correctly, because between Windows and/or Linux x86 systems you don't need any of the "transfer text between dissimilar systems" that ASCII mode achieves. ASCII mode can only get you into trouble or create unexpected results for these cases.
I haven't tried to transfer anything in actual ASCII mode in quite some time; I wonder if maybe that's part of the conditions for the symptoms you're observing. These aren't the symptoms I would expect (it would usually be corrupt files, as you experienced earlier), but maybe the particular FTP server has a unique reaction to the ASCII transfer. You might try switching to binary-only and see if the issue even continues to exist.
Filezilla lets you enable verbose logging in the "Debug" section of Settings. The additional info is going to appear in the normal log window. If the issue still occurs, I'd start with a known-empty directory and then focus on one of the files that prompted for overwrite, and see what actions occurred related to that file within the debug log.