No, it's a masculine noun, so it's "kjæresten" regardless of the gender of the person.
Why does "kjæreste" translate to "sweetheart" and "partner" but not "beloved"?
Beloved would translate into 'elskede'.
But I would say that 'kjæreste' has so many different meanings that 'beloved' could also be used.
In informal contexts, most (particularly young people) will use "Typen" (The guy), and "Dama" (The lady/The Dame). Neither is a super affectionate term, and most of the time is used in the posessive context, "typen hennes", "Dama hans". Addressing your own partner with either is at least a bit of a sleight. For affectionate terms, "elskling", "elskede", both lit. mean beloved, but is gender neutral.