We just had a Trane R-13 outside unit installed two weeks
Ago. While it is running water drips out of the bottom corner of the unit.
The installer says this is condensation, and it will do that when it is
New, but it is still doing it a few weeks later. We have never had any
Water come out of an outdoor unit before. Is this normal, or is the unit
Defective?

Condensation will form on the refrigerant lines and compressor, this is more prevalent if the system is overcharged, the humidity in your local area makes a difference in the amount of condensation.

Humidity is high in our area (DC) however our other unit (also a trane) does not have this problem - my wife is worried that the unit is defective since the water drips continuously as it runs. Any thoughts?

Look at the compressor and see if it is really wet, this might indicate an overcharge of refrigerant, also make sure the filter is clean and all the registers are open. Now measure the air temperature across the evaporator coil, this may be dirty and restricting air flow, which can cause the system to run colder and sweat at the compressor. The temp. should be 20 degrees, if higher then this it means not enough air flow, the slower the air flow the greater the differential.(Air flow problems)