That sounds stressful, but you might as well get used to it -- the price of being competent is that responsibility accumulates around you.

One suggestion: keep track of the work you've done and what you have been asked to do, and make that record public.

This could be an RT system or simple database, or it could just be a clipboard on your desk with boxes to write in the name of each task, who asked you to work on it, and how much time you've spent on it so far.

This serves a number of purposes:

• It acts as a paper trail, so that if your supervisor asks you at the end of the month "what have you been spending your time on" you have something to show them, rather than just waving your hands.
• It makes your work queue visible to people who are making requests, so that if someone asks you "could you get this done by tomorrow" you can point to the list and say "yes, but only by making all of these people wait longer for the things they've already asked me for."
• It helps you get a plausible sense of how much work you can get done each week, so that you can give people more realistic estimates of when you will finish their task.

It's not a complete solution, but it can help provide a little structural support as you adjust to being in demand.