[LMP]

Obviously, the battery is taking all the load.
Yes...alternator is first suspect...connect a voltmeter to the alternator (heavy gauge wire at the back) , run the engine...should read around 14.5 volt.....then test at battery (engine still running) ...should read the same. (or the other way around) . THere is a fusible link between the alternator and battery. Failure of this link will not charge the battery even if the alternator tries.
IF the battery stays at 12V (engine still running and alternator showing 14.5 v) , the fusible link between alternator and battery link is shot.

OF course, if the running alternator stays at 12V (which is battery voltage carried to the alternator) , the alternator is shot....in most cases, the built in voltage regulator ($20-$30) is the culprit and it can be replaced if you or a friend can use a soldering gun. I did it twice...( .I also changed the built in diode bridge once: one failed diode (one of six) reduces charging power...but not entirely).


Testing for the fusible link can be done like this also: Test the heavy gauge wire at the back of the alternator with engine not running and everything off. A voltage test is a first step...as reading zero at the alternator would indicate link failure, period.......but even if it shows the 12V battery voltage....it can be misleading as a failed fusible could carry enough of that tiny current (from the battery) that is sufficient for the voltmeter to show a voltage....so use something like connecting a headlight bulb to this wire on one side, and ground on the other ...if the light stays dead, the link is shot.

Sorry you change the battery..unless it was already 5-6 year old (normal lifespan) which would then be considered "preventive maintenance".. Batteries are often the victim, not the cause, of other failures...

...reading back your post, it seems however that some charging occurs when engine runs at higher speed...as you say "lights dim when engine idles"..... are you sure nothing is draining the battery while car is stopped ?