This article from 3-15-06 announced Lake as host. I don't know when the sets were built, how long after the announcement that the shows taped or what kind of time frame they were all taped in (Feud, being the finale, would have been the last to tape). I don't know when Feud Live started, but that thread with the pics was from 4-10-06. So, as far as which board was made first, it would pretty much be FF Live's (so maybe GSM's was based off of those plans). I don't know how long the Live show originally ran, but yeah, chances are they were going at the same time.
I was just killing the theory that Live "re-purposed" the GSM set when (to me) if any "re-purposing" went on it would have to be the other way around (GSM took the board from Live) if Live was done/on hiatus by then. Either that or they borrowed it (if Live was still running when GSM taped, but that seems like a lot of work to transport and change the border and such not counting that if the show was running they need the board) or they built another board. Both boards do look pretty identical....