Keep in mind you're never going to have a perfect system - call me a defeatist, but you're never going to get the repeat question problem to drop to a point where it's "gone". So first maybe it's good to get a very cold look at the situation and see "how bad is it really?" If the issue is just that the response to the same question being asked is the same copy/paste, it may not be an issue worth chasing down too much.
Information dissemination can be very tough, because a system that works for one person doesn't work for another, so two otherwise equally skilled persons may not find the same information in the same manner. It's very tempting to think you can engineer away this problem, but it introduces many more.
If you have repeat offenders who really are eating up a lot of time for everyone (I'm not getting from your description if other CS also participate in providing answers or if only Engineers do), then it's worth having an educational talk with them. "What did you try checking first? What search terms did you use? What information did you get on the problem first before coming to the channel to ask?" Often times the problem isn't the process in the information store (i.e., your FAQ and Engineers + more experiences CS reps), but in the way that the information is being processed by the CS rep, and how they try to feed that information in their available sources.
To clarify on the sources bit, different systems treat search differently, adding weight to different parts of articles and entries, etc. We switch our internal wiki system over a few months back, and it took awhile to get used to the search on the new system; it offered more granular search arguments, but people had gotten used to the relatively simple Title+Document Content > Title > Document Content search on our previous system. We had to retrain people a lot on how to "search" for information, and also had to really take time to structure the content in our new system to be discoverable by how our team was searching.
So, in short, figure out how big the issue really is. It's never going to be perfect, or even close. For your repeat questions/offenders, investigate - see what the people tried before hand, and figure out if it's an issue with what the CS reps are gathering/presenting or of there is an issue with discoverability in your FAQ/other information systems.