Spinning at a high cadence seems to be one of those things that's benefits are grossly overstated and actually there is a lot of empirical evidence showing that cyclists are often most efficient at a cadence well below what they believe to be their most efficient.

It's all about finding what works and is comfortable for you. From what you say about tiring out quickly, it sounds like you just go too hard for your current fitness level which may or may not have anything to do with cadence. I was very similar when I got into cycling, and in pretty much all other athletic endeavors, basically going as hard as I can for as long as I can. What I've learned is to better understand what I can sustain for different lengths of time. The Zone 2 riding that has been mentioned by others is indeed a great way to build your base fitness which will ultimately enable you to ride faster for longer. If you don't have one, I'd recommend a heart rate monitor to give you instant feedback as your riding as to exactly how hard you are working. My guess is that you'll see that as you currently ride, your heart ride gradually rises as you maintain effort through your ride, indicating that you're riding above an effort level you can maintain.

It'll take some conscious effort, but if you read up a little on FTP and heart rate training zones and can work to back off the effort slightly to keep your heart rate from continuing to climb, you'll gain performance sooner and be able to maintain speed more easily. This won't necessarily require higher cadence, but that's often the easiest method to start with. Have a goal when you ride of what range you want your heart rate to stay in and when you start creeping toward the high end, drop a gear or two and try to keep from going beyond that number. Once you do this a few times, you'll start equating how you feel to where your heart rate is and it'll take less frequent monitoring to maintain a more consistent effort. Also, more conscious of incline and change gears as it changes. You may be able to power through uphill sections but what is probably happening is that those sections are raising your heart rate and it's not recovering down and then the next one raises it a bit more and so on, until you're getting beyond what you can sustain and that's when you're getting tired. Backing off on the gears and accepting that it's ok to go slower on the inclines will keep your effort more consistent and prevent fatigue from setting in as quickly.