"Balzende", which comes from the noun "die Balz" (verb "balzen"), usually refers to animal courtship. It is hardly ever used for humans in the sense of "courtship", except as a derogatory expression, if you want to emphasize the animalistic part of it (e.g. you might refer to a gang of horny teenagers trying to get some girls' attention as "balzende Kinder"), but even that is rarely done. Also, this really only refers to the act of "courting" before any kind of relationship (even if it is only sexual).
The better translation for "courting" would be "Werbung", but I guess this one is a bit antiquated nowadays. However, also this refers to everything before and maybe including the dates until some form of relationship is established, not really anything more.
Guntram already gave an answer of what you could call "dating" in Germany (and I'll also take his definition as a basis), which basically sums it up. However, one should be aware that "die Beiden sind zusammen" is also used for serious relationships, while "die Beiden gehen miteinander" in my experience is more used for teenage relationships (it's the cliché phrase for notes passed between I don't know, maybe primary or middle school pupils: "Willst du mit mir gehn? Ja, Nein, Vielleicht").
An alternative might be "Die beiden gehen aus" or "Die beiden daten" - however I'd say this might be less than what you want to say with "dating". That said, a lot of terms revolving around not so serious relationships (e.g. friends with benefits, one-night stands) are understood in German and do not have an exact translation.