I'm the kind of girl that'll swallow every line - meaning?
I'm the kind of girl that'll swallow every line
But it's all right
You only gotta call
And I'll be there any time
And it's all right

Lyrics from by Dooleys.
Here, "swallow every line", does this mean
swallow her lines or his lines?
In other words, does this imply
she's so introverted that she couldn't even speak out her feelings and just bear with it,
or
she's gullible to the point that she would take all the sweet talks seriously?

Question

I'm the kind of girl that'll swallow every line -> meaning?
> I'm the kind of girl that'll swallow every line
> But it's all right
> You only gotta call
> And I'll be there any time
> And it's all right

Lyrics from  by Dooleys.
Here, "swallow every line", does this mean
swallow her lines or his lines?
In other words, does this imply
she's so introverted that she couldn't even speak out her feelings and just bear with it,
or
she's gullible to the point that she would take all the sweet talks seriously?

Answer

Your latter guess was correct. She is saying that she is gullible and will believe anything she is told. To "feed someone a line"  is to speak disingenuously to that person. You had the correct idea!