Probably not a good idea to disconnect the battery with the engine running. Alternator output can go very high and not only can they self-destruct, they can take out other electronics, such as your expensive ignition computer.
Here's some info:
http://www.aa1car.com/library/charging_checks.htm
ps: I guess it can't be the problem in your case as you've been driving so long, but sometimes alternators are shipped with the brush hold-back wire still installed. They shouldnt be as all alternators should be bench checked before shipping and the wire would cause the test to fail. It's a little wire that is temporarily inserted to hold the brushes back until the armature is installed. Goes in a hole on the back of the housing. Then the wire is supposed to be pulled out. Also, some manufacurers put a shorting stip from a ground stud to the output terminal to keep static electricity from damaging the diodes. Something that has to be removed before use.
pps: Biggest problem I've had with alternators is dirty contacts. Dismantle and clean usually fixes things up. Second most common problem is fried diodes.