Welcome to the Asana Community Forum @Gosia08
Here is a great overview/graphic on the project access levels:
Now you mentioned you are on a trial version, did you main Premium?
Then the comment-only access is available. More info here.
Now if a task in a comment-only project is also added to another project that has no restrictions, to which the same person has higher access, then they would still be able to edit tasks.
For a task within a project where somebody has comment-only access, they will not be able to amend due dates, except for tasks assigned to them.
Any task assigned to a person shows up in their My Tasks also and they will be able to amend the due date (even if they have comment-only access in a specific project)