I sense you feel you're doing something wrong, or maybe not doing as well as you could be, so first a thought which may make you feel better...you're overworked. Many studies have shown that the proper ratio of IT support people to total employees is anywhere from 1/20 to 1/50 depending on environment stability and various other factors. Irrespective of whatever's going on in your shop, you don't have enough people.
With that as context here's some advice on how to make things better:
1. Prioritize. You can't do it all. You can't sacrifice the urgent for the important. Most importantly, you can't be the one to figure out the priorities...that's not IT's job. If you don't have some kind of a user group, start one. Get your business customers to prioritize your workload, but always in context...not everything can be "A" priority. Make them build compelling business cases for the high priority items, and make them justify why the low priority items should even stay on the list.
2. Decline. Just because you're asked to do something doesn't make it a mandate from God. There are only 24 hours in a day and you're entitled to some of them. When people come up to you with their hair on fire and tell you the world will come to an end unless you rebuild your network topology and have it done first thing in the morning, don't succumb to panic and assume they know what they're talking about just because, well, they must. Negotiate timeframes and scope that make business sense and are achievable.
3. Communicate. There's never enough of this. Make sure everyone who needs to knows what you're doing, why you're doing it, how far along you are, and when you'll be done. Be responsible and responsive. Be accountable. Be visible.
4. Execute. Nothing succeeds like success. Build up your credibility by getting the important stuff done, and when your customers all start swarming around you like sharks at a feeding frenzy you'll have the confidence and the track record to be able to push back.
5. Meditate. I'm not suggesting you join an ashram or light incense in the LAN room, but it's OK to take ten minutes once or twice throughout the day and do some controlled breathing exercises. Very refreshing, very calming, and ultimately very productive.
Just be the best professional you can be. No one can expect more than that. Revel in your successes and don't worry about the failures. View everything in the long term...when you factor in the dimension of time, everything works out in the end.