I would say you should try spinning in a lower gear. A higher cadence is is easier on the joints and muscles and really allows them to work longer. I think the tradeoff is that is relier on your aerobic system more, but it seems that your muscles are dead before your lungs are. I suppose that makes sense.
Try to maintain a cadence of at least 90, up to 120. It is easy to know if your cadence is 120 if you have a watch because your foot makes 2 full circles per second.
Also, focus on keeping your knees in line with your feet (this usually feels like your knees are pushing in) and having a smooth pedal stroke where you are spinning in 'perfect circles', and it feels like you are putting power down the whole time, not just in up/down pulses.