Taylor Sweet a-running to sweet nothings...

They said the end is comin'
Everyone's up to somethin'
I found myself a-runnin' home to your sweet nothings

Dizem que o fim está chegando
Todo mundo está tramando algo
Eu me vi correndo para casa, para suas palavras doces.

A-running is "correndo" but has a ring of "na hora", you run with energy, that she have "an incentive"
To run, Having a strong urge to run and to arrive to the place, etc.

A-verb gives that emphasis of "fazer algo corendo" as in the example:
She refused to come over for dinner, but she came a-calling when she needed someone to feed her cat for the weekend.

Ela recusou-se a vir jantar, mas ligou (ocrrendo) quando precisou de alguém para alimentar o gato durante o fim de semana.

Many people would say it's old-fashioned, but if Taylor Swift sings it...

Ou "ligou rapidinho"...

Ref. forum.wordreference

WordHippo describe "sweet nothings" as :
Insubstantial or romantic words that are only meant to flatter, woo, or seduce.
Another definition is "talk or praise used to charm, flatter, or persuade". Synonym with "honeyed words/romantic words.