Excusivity if Redweek posts my TS for sale?

I am trying to understand if by having Redweek post my timeshare for sale I am locking into an excusivity agreement with them or I can post with other sites and entities.

No, there is no exclusivity. You're free to post your unit(s) for sale or rent on other sites.

You say your are "locking into an excusivity [sic] agreement with them", does it stipulate that in your contract? It might if you are using RedWeek's Full Service. However, if you are just using RedWeek's listing service, then there should be no exclusivity.