Looking for the best task management software or method to fit my unorganized brain

I have been struggling lately with managing all of my tasks at work and outlook reminders and to-do aren't cutting it.

In my job I deal with the needs of 1000 employees that all have a variety of requests/needs some are low priority and can wait a few hours or days, others are drop what are doing at get this done now, others are long term projects that I am waiting on something from another party but as soon as I get that I will need to get back to that project as soon as possible.

I was wondering if their is a good task management or even PM software that would help me track and very visually prioritize my tasks.

Ideally I would like something that tracks specific tasks based on a group I assign it to and then also has a dashboard that summarizes any tasks that are due ASAP or overdue.

My IT department isn't a fan of linking company mail to external services like Monday or Notion. (In fact the completly blocked notion from our network). I really like Notion because of how flexible it is. Ideally I would like a system that tracks different task/case types and has an overview dashboard of any tasks that are overdue or considered high priority. I am in HR so examples would be like Here is a dashboard of your open FMLA cases and at the top are the ones with a due date of today or are at a set threshold for following up. Then a separate dashboard for reoccurring tasks I need to do each week. I hope that gets what I am looking for across.

My last job used Salesforce CRM as a case/task management software and I absolutely loved it. My current company isn't willing to explore that as an option.

I've been frantically searching the web for ideas but have not had much luck or it's difficult to tell if a certain method meets my requirements.

TLDR; looking for task management/project management organization advice for a role with constantly shifting priorities (be that software methods or a more physical method.)