Thanks again for for the suggestions and comments.
I bought a multimeter and now can report more information that may lead to a solution for the quick draining battery in my 37E.
When leaving the battery to charge overnight, I receive a respectable 2.6V reading. After removing the AC adapter and turning the calculator on the low battery indicator turns on after three minutes. A few of the LEDs start to dim or flicker at five minutes (isn't this dramatic?). By seven minutes the display is dead. I then check the voltage of the battery and it registers 1.7V.
The 37E funtions perfectly on the AC adapter.
Other potential information that may be helpful in the diagnosis is the voltage at the cord is 11V and the power at the battery terminals is ~7.2V.
This may be unrelated, but I do sense that there is some play in the off/on switch.
Your thoughts and suggestions will be greatly appreciated.
Dan