There is a bug that can appear on EVOs. When the battery drops below a certain point, the phone will no longer charge it. Sprint and HTC both know about it and have said it will be in the next MR.
You can pop the battery on a standalone charger for 20 minutes, then put it back in your phone and it'll charge fine. Until the MR is released, techs really should be exchanging these phone.
Not saying this is your problem OP, but it might be and is something else to make sure a tech checks. You could also order some dirt cheap batteries and standalone chargers off ebay. Try to keep your good battery juiced. If you forget and can't charge it, swap to another while you put it in the standalone. Then, just wait it out. A fix will be coming for sure. If this is the problem, it's software issues, not a defect in your phone.