@Oliviahuffman Are you still getting the "connecting" message at the Main Menu if you do NOT try to login again? If so, please clear the Sims 3-related Keychain Access data again and try without signing in, which is not necessary to play.
If this doesn't help, please move the Sims 3 folder in Documents > Electronic Arts to your desktop, delete the Keychain Access data (for the third time), and let me know what happens.
Can you sign into an EA website with your current username and password? Please test and let me know as well.
By the way, the launcher and game often don't accept passwords with special characters in them, for example & or !, so if yours has one, please change it.
