This is something that takes some practice to feel comfortable. As you shift to a harder gear your cadence will slow and your speed will increase as you mentioned. That is normal since your putting more power into the pedal stroke. Find a gear that allows you to pedal at a cadence that isn't too fast, but allows you to stand and stretch.

You could try the following interval: at every 5 minute interval, stand and pedal for 30 seconds. As this becomes easier extend the standing pedal to 1 minute, and so on until you can to 2-3 minutes without fatiguing. It will get easier with practice and is good to stretch out the back and legs on a longer ride.