How to organize the work of the department of programmers in a state institution with a hectic flow of tasks?

Question

Work organization

My question is an attempt to find an option for a good organization of the work of the development department in a state institution.
Given:
Large university (about 5,000 employees and 20,000 students),
Web development department of 6 people (I have been the head of the department for less than a year)
Responsibilities include: self-written website of the organization and various internal information web systems, technical and user support for these systems, as well as rewriting all systems from the old platform (Orcale) to modern ones.
Problems:

• Most of the tasks come to the department in the form of orders to "do this and that." That is, we can only influence the details of the implementation.
  
• Due to some bureaucracy in the organization, some tasks come to the department late, when there is not much time for feedback - you have to take it and do it, trying to improve / simplify something on the fly.
  
• For a large part of the tasks, there is no acceptance, as a result, the department has to take on both the acceptance and correction of tasks. In this regard, there is a suspicion that some tasks do not do what the institution really needs, but what seems to be necessary.
  
• Due to the frequent adjustments of tasks by departments adjacent to our department, which will then use our systems, the department jumps from one task to another, which reduces productivity and prevents them from developing systems - obviously bad.
  

Question: is it possible to build communism in a single country? That is, is it possible to more or less regulate the interaction of the development department with customers within the organization? Maybe someone has a similar experience? Or at least in the direction of what buzzwords and books is it worth digging?
So far succeeded:

• somewhat slow down the flow of applications by switching from telephone communication with subcontractors to email, which made it possible to less frequently switch the existing two and a half programmers between tasks.
  
• Within the department, I organized a public queue of tasks (something like a kanban) so that employees better understand what is happening.
  
• There is a desire to outsource small tasks unrelated to the core of the system.
  
• Now I'm preoccupied with finalizing the regulation on the department, but I don't quite understand how to formulate all this in it.
  

I would be grateful for any thoughts on the topic.

Answer 1

You are doing everything right, well done!
organize a rotation of student slaves (practice or working off - xs what they need now, maybe the first experience for a resume) - this will slightly reduce the
acceptance load - it’s customary in the academic environment to defend one’s F and no one will be substituted because of computer scientists
as much as possible - formalize setting tasks (it is unlikely that you will achieve this from the administration, but from the departments - completely), this will detach any left-wing ball earners

Answer 2

1. Instead of an email, start a service desk system (JIRA, or something free), through which tasks should actually come. If some "customers" are not capable, let them send emails or documents that can be attached/embedded from them into JIRA tickets.
It will be possible to manage priorities, it will be much easier to submit statistics to the top than people were busy with, give an example of the cost in hours to solve a problem or to clarify it in order to identify for management which of the customers is not able to set the task normally, which causes problems.
2. When the statistics are collected, go with it to the top management and seek influence on the "orders".

Answer 3

Implement project management.

Answer 4

Read Target and Project Phoenix.

Answer 5

Scrum

Answer 6

Theory of Constraints of Systems E.Goldratt

Answer 7

"Due to some bureaucracy in the organization, some tasks arrive late to the department"
Where do the tasks come from? Who puts them up? Do you communicate with them?
If everything is bad, tasks cannot be avoided, you can at least establish relationships with people working in the department / subdivision from above and setting tasks. So you will receive copies of the tasks on the soap before you get the paper version in the form of a letter / order.