The sentence that correctly used the word "sway" is sentence 2, "Even though Beth was too shy to talk to Ben, Ben's feelings did not sway.
The word sway has a couple of definitions:
- to move back and forth from side to side
- to control or influence (a person or course of action)
Sentence 1 is almost correct, but should be reworded to correctly use the verb, like seen below:
1. Manuel is going to sway the club to start accepting new members.
The other sentences need a bit of revising if the verb "sway" could be used in them.