It appears your priorities are changing far too frequently for either methodology Agile or Waterfall. With priorities changing frequently, you are likely churning in and out of projects leaving a lot of them partly done. The Agile alway be ready to release may help. It has been my experience that getting a methodology in place will improve productivity.
Your situation reminds me of a project I worked on. The developer on the project asked one question at the start, "Do you want me to be do it right or be responsive?" I was on the project when it was two years into a six month project. One week the same functionality was implemented Monday, Wednesday, and Friday. Tuesday and Thursday were spent removing the functionality.
I would suggest you start adopting practices from Agile. Scheduling a short sprint period could help with changing priorites. It may be easier to maintain priorities for a period of a week or two and may make it easier to stabilize priorities. You will also need a backlog (sounds like you have a large one already).
Management may be more willing to hold off new priorities if you can slot them into a sprint in a week or two. You will also be able to identify the priority tradeoffs. If you add something to the next sprint, what will be removed.
Consider having part of the team working Agile while the others maintain the status quo. Rotate one team member each sprint as you are gaining experience. Consider having the whole team participate in a daily stand up status meeting, and the post sprint review. Once you have demonstrated increased productivity and returns to the company you should be able to increase the amount of work being done using your methodology.
Agile is a adaptive methology. Expect to be making major changes to your methodoly for the new year or two. Eventually, you should reach a stage where you are fine tuning.