What model of alternator do you have, and is your car equipped with an external voltage regulator? It sounds like you have the field lead connected to an unswitched circuit, which WILL drain your battery.

The hot/slow cranking is a separate issue, quite common, due to heat and degradation of the starter, solenoid, and/or battery cables.

See http://www.hartin.com/alt.htm for some basic information. See http://thefirstgensite.com/library/electrical/68wir/68wir1.html for the althernator wiring diagram for a 1968 Firebird.