Many, many, many moons ago I had a job in a production environment that required fast turnaround of "designs". I use quotes because in reality it was far more typesetting and layout or production than it was actual design.

This was really an invaluable position for my career as a whole. It gave me a strong footing in production which has only served me well over the years. So, even though it may be more production than design, there's a great deal to be said for spending a few years in that type of environment.

When I was working in this position I customarily spit out 70-100 projects daily. (no exaggeration, I kept track) Granted many of these were simple "put a name on a business card template" or things of that nature. But there was a mountain of work and only myself to go through it all. This was before the days of the internet, back with 8MB of RAM was HUGE (yes MEGAbytes). So... it was quite a while ago.

What I learned to assist in organization....

1. EVERY job must have a job ticket describing what is required. These tickets are to be filled out by the coworker requesting the work. Incomplete or unclear tickets would merely delay production and delays were understood to be due to the coworker, not me. This served to get accurate, clear job tickets most of the time.
2. Job tickets were stacked as first received, first completed, no exceptions. New tickets were placed at the bottom of the stack.
3. I, personally, never jumped the stack. Whatever was on top... is what I worked on next no matter what it was or what I "felt" like working on. The top job ticket was the next to get done in all cases.
4. In my position I had 6 people filing job tickets. I provided each of those people with only 2 "rush" opportunities per week. Yes, you'll repeatedly get requests to "jump the line" but all that does is cause pile ups. So by providing 2 chances to request a rush, they used them sparingly rather than for EVERY ticket.
5. Errors/corrections due to my mistakes were always treated as rush projects and completed "next". So, there was never more than a day's delay with correction. These types of errors were most commonly things like typos or misalignments - quick easy fixes.
6. Errors/corrections due to a faulty job ticket were always completed by 10am the next day. So there was never that much of a delay with those either. But these errors probably entail more than a quick typo fix.

Beyond managing the inflow of work, keeping a well organized file structure on the computer is also imperative. How you organize things is not nearly as important as how intuitive the organization is to you. You need to ensure you use file and directory naming conventions that quickly allow you to find things. Don't title all business card files "BusCard1.indd" that's not helpful. Use unique identifiers in all file names - company name + file type (i.e. McDonalds_BusinessCard.indd) or whatever, just so long as it's not generic. This allows you to search more accurately and to identify file contents merely by the file title.

In today's world.. I'd imagine most of the job tickets I had would now be emails. So, I'd set up a specific email address for only job tickets and configure a few inbox folders... New, Completed, Delayed, Rush. All new ticket emails would file in the New folder, then be moved to others accordingly. Sort the New director in Descending order and whatever email is on top of the list is the next project to complete.

I don't know that any of this will help you specifically. It's just how I handled things. One important thing I learned is to speak up if you feel you are being pushed too much. No one will just decide to give you an extra day for something. Most likely everything you touch is wanted YESTERDAY!!!!!!!!. So it's up to you to set expectations. Get people used to 2 day turnaround on new tickets due to your work load, and 1 day on edits, corrections, or updates ... then eventually you may be able to get them to adjust to 3 day turnaround for new tickets. This will provide you with a little breathing room. Truth of the matter is I still handle things this way today.. new projects are always set with several days or weeks to initial delivery -- while edits, corrections, or updates to existing projects are always completed as soon as possible.