As the responsible engineer, I can't mark my task as done and see that on the project list

[Ron DeSpain]

When the DB manager marks the task done it disappears from the list, but until it disappears I can't tell if I've fixed the issue without opening each issue and reading the comments.  A status needs to be available in the task list that is set by the responsible engineer.  Otherwise the engineer must maintain another database with that info, and it's almost impossible to keep two independent databases synchronized.

[Ron DeSpain]

I'm not the dB manager, so I don't have access to the resolution field.  I've figured out a way, by adding text to the project name "Done rrd" which shows up in the system dashboard.

[Nic Brough -Adaptavist-]

Everyone can see the resolution.  You may not be able to set it due to workflow, but it is visible in the issue view, no matter who you are.

It sounds to me like you have boards or views that are set to ignore resolved issues.  I would get them changed so that they include them.  Or just create your own ones that work for you.

[Jack Brickey]

where are you viewing the issues? if you view them as cards on a kanban/sprint board they will have a line thru the issuekey if the Resolution field is set. if a filtered list the Resolution field can illustrate this.