... and then use Seasons' new password on another save file on your Ages cartridge? I can't seem find this information by searching online. I'm just wondering because I'd like to get the "full experience," as it's my understanding that using end-game passwords on the games add special content unique to each cartridge. This would mean I'd only have to play through the stories three times (Ages, Seasons, and then Ages) instead of four (Ages, Seasons, Seasons, Ages).

I think you can find the passwords online anyway, but yeah I believe you can use the codes like that.
 
EDIT: Codes are available online, but they restrict your name. Using skills of deduction, I believe you can, but I'm not sure if it's the best idea, because you'll miss out on linked Ages content AFAIK. It's been forever since I've played Ages, which was the only one I had, so...yeah...

Beat Ages, get a code
Beat extended Seasons, get hero code for Ages
The hero code gives you an extra heart and all the benefits of a password linked game minus the extra story content.
So to experience everything without looking up codes you would have to beat both games twice.

The difference in extra story between the games can be experienced in a couple screen shots.
I just played through both a couple weeks ago.