You can run the Garmin Edges (including the Touring) while they are being charged.
The OP's problem seems to be the result of not using a suitable charger (most chargers will work).
Actually, if you read the update, the initial unit was faulty. The replacement will shut off if one attempts to charge it when it's all the way charged already, but if one doesn't attempt that, it's fine. I'm using the standard charger everyone recommends (at least among AA-based chargers) -- the Gomadic.
Let's not beat a dead horse, please; I just wanted to clarify since BF comes up high in search results, and if anyone else runs into issues and finds this thread, I'd like them to see the whole story.