I'm not an IT person, but I'd suggest, before you go to management, you optimize the department.
You mention that you are taking on projects you don't have a chance to even start. Also you talk about mundane tasks preventing you from working on other projects. You should make sure that, given your current resources, you are working on the right tasks. Start saying no to the least important things. Only take on tasks that you know you can complete. If what you say is true then thats already a big improvement.
Think carefully about this, and discuss it with your manager and other departments before you implement, because you are going to get a lot of pushback. Your approach should be "we want to start focusing on critical work, we don't have the resource to work on your project now". Absolutely, don't use it as a play for more staff or you'll find yourself in trouble.
Only go to management with a request for more staff when you can show a benefit to the company. A list of additional tasks that you could take on and what their impact would be for example.