I am going to be a real jerk here and give you the guitar teacher answer, which is not one you'll like--You may not be holding your hands or the guitar properly, and the Baby Taylor is OK to play simply because the strings are too light--

If your hand is cramping on the Alvarez, you should   reconfigure your wrist, arm, and fingers until you can do what you need to do without cramping--If your technique is good, you should be able to play basic stuff on most guitars easily (though you always have to adjust a little to accomodate the differences)--