Tube damage around the valve stem is often caused by rough handling when pumping or connecting/disconnecting the pump head. I went through a series of tubes the same way years ago and I was sure it was faulty tubes, but the problem stopped soon after I learned to support the mini pump from the rear while pumping.
What kind of pump do you have?
Also, 60psi is 100% acceptable in most 26X2 or 26X1.9 tires if you are riding on the road.