It has been a while since I have looked into ticket tracking (call tracking) software, but I was playing with some years ago. There were a few free options, all web-based, back then. I didn't use them because it meant having 2 tickets for everything I was working on, one in the accounting software for actual invoicing.
The software I was looking at was really nice. I have a bad memory and I wanted to keep track of a lot of information. First, I wanted to keep track of interactions with customers. If I get a few "hints" about something it all seems to come back to me. I just have to relate the face I'm looking at or voice I'm hearing to one specific situation and then I remember who they are. So this software would allow me to take notes on calls and interactions as a memory aid to who a specific person was. I have this thing where I associate names or faces with places instead of each other. Just yesterday I was working on a computer and the girl had given me her mom's number to call. The name sounded familiar. On the phone I didn't think much of it. Then I looked at the picture on her background. She calls me often. I go to her work site often. I know her as well as I know any client, just not her name, voice or the old name for her place of employment, where she has been for over a decade but which has changed names.
The second thing I wanted to do was take notes as I was fixing things. Many times I have found fixes to weird issues, but can't remember them when I need them again. This software made a searchable database, so I could just do a search to find what I had done in the past for a particular issue.
This software, as I recalled, was really a "call tracking" software. But it let me create tickets, assign them to a user, assign them to a customer and, eventually, close them out. I think it even had the ability to allow a "manager" to approve closing out a ticket. My memory isn't great, but as I recall it sounds like it did everything you wanted. And at the time there were a few free or "free-ish" web-based solutions out there where all you had to do was set up Apache on a server to handle it. As I recall, there was a paid version with more functionality. Try searching "ticket tracking software" and, especially, "call tracking software". As I recall, the search was tedious and time consuming, but, starting there, you could eventually come up with maybe as many as half a dozen free solutions. Keep in mind this was about 15 years ago. I have looked again from time to time over the years, but never again to that extent.