This is a note I’m leaving for my future self if I have this issue again but hopefully it will help anyone else experiencing the issues I was.
Basically, what was happening is even though I was using the right sign-in credentials for my google account with the Sparrow and Adium apps, both of them kept re-prompting me for the credentials. However, this was only happening when signing in from home. Elsewhere, there were no problems. Trying to find the answer via google took a lot of searching but the solution that worked for me was to just go to this web address: https://accounts.google.com/b/0/DisplayUnlockCaptchaV2
Apparently, at some point Google locked out sign-in attempts from those apps (and ip addresss) because of too many unsuccessful attempts – its a security feature (although not widely published or known). Visiting that url temporarily deactivates the security lock and then you can sign in.
It worked for me and everything is back to normal thankfully!