Just because you put a new alternator in doesn't mean you didn't get a bad one. With your car running, put a voltmeter (or a multimeter set to volts) on your battery terminals to see what voltage your getting. A cars battery should be charging at 13.5-13.8 volts. The dimming headlights makes it sound like your having an alternator/charging issue.
That or you have whats known as a "phantom draw". Something may be draining too much power while your car is sitting overnight and it's wearing the battery down to the point that it can just start the car.
How long do you have to crank before the car starts?
1966 Mustang Sprint 200--Wimbledon White
1998 Mustang V6 Coupe--Silver--RIP, Totalled by brother
2005 Mustang V6 Premium Coupe--Sonic Blue