The duo will sign two-year deals ahead of the club's first season in the Championship following promotion via the play-offs last month.
Foley joined the Glovers from Newport last summer and made 41 appearances, scoring five goals.
Dawson signed for the Somerset club in January after his contract at Shelbourne expired. He made his debut as a substitute in a Johnstone's Paint Trophy defeat to Leyton Orient.
Dawson went on to make 20 appearances last season, scoring a vital goal against Sheffield United in the play-off semi-final to help Yeovil on their way to Wembley, where they would go on to gain promotion against Brentford.