She had no idea if seeing the name would have inspired this version of her with happy anticipation or sinking dread.

[shootingstar]

Nora is dreaming of other selfs. In this context she ist getting in touch with the new name "Nadia" she hasn't known or heard of up to now; in her youth Nora was an excellent swimmer.
(Nora's other version of her is talking on a video on her phone. Nora is watching it)
'. . . Work hard at it. And don't give a second thought when people mock it or ridicule it. Most gossip is envy in disguise. . . Keep swimming . . .
'Keep swimming,' Nora mumbled, echoing this other self and wondering if the hotel had a pool.
The video disappeared and a second later her phone started to buzz.
A name appeared. 'Nadia'.
She didn't know any Nadias in her original life. She had no idea if seeing the name would have inspired this version of her with happy anticipation or sinking dread.

(The Midnight Library by Matt Haig, episode The Successful Life))

What does the word "sinking" mean in this context?

 

[emsr2d2]

It refers to the feeling you sometimes get in your chest or abdomen when you're scared or worried. It feels a bit like you're falling, or the feeling you get on a rollercoaster or when you go over the top of a high bump in the road.