It sounds like you really have a dovecot problem, not a gnus problem.
dovecot needs to be able to write to your mailstore.

I would suggest pointing some other imap client against it and seeing
what's happens.

usually people start dovecot as root, have some kind of user/pass
login, and  dovecot setuid's to the use.

I have been using gnus with dovecot for a very long time, maybe 2005 or
2006, with no permissions issues.