My apologies to those who said my problem had to do with a failing battery. The fact that the battery went south the same time as I installed 10.3.6 was an interesting coincidence that mislead me.
I first tried all the other suggestions I received and none of them solved the problem so I ordered a new battery. That did it. I observed one slight problem. The amber charge light only seems to be on when somebody is logged in. If nobody is logged in or the computer is turned off it is green. This seems to slow down charging but eventually the battery does get fully charged.