Congratulations, even though you’ve found yourself in a tough situation. As a new graduate, you still have lots to learn about the practical, day-to-day parts of the business that were likely never addressed in school. It sounds like you’re running headfirst into those problems.
One thing to consider is that developing solutions to design problems takes time; it doesn’t just happen. You might not be to the point where you can accurately estimate the time something takes, but you need to make your best guesses and prioritize your working hours to get the most important things done. If the workload is too great, honestly, you can’t work past your limits — nobody can. While you’re prioritizing and scheduling, don’t forget to factor in the time it takes for research, thinking, revisions, etc. Design projects are rarely straight-forward — especially when you’re working with other non-designers.
PrintDriver made a very good point about thinking through the bigger problem, the solution and its implementation — all the way to the end. Don’t dive blindly into design problems. Think it through to its conclusion before you start — especially the practical stuff that designers sometimes tend to ignore. You might change course as you’re working due to detours and finding better routes, but that’s OK — at least you’ll start out with a clear direction and destination.