I picked up my camera the other day and it wasn't working, the screen will not turn on no matter what button I push. But today it did say to replace the battery. When I turn it on the little red light on the side blinks. If I look through the view finder it numbers will flash. I click the buttons and it will focus but then nothing.
I don't think the T3 has a separate clock battery. They generally have a capapcitor that maintains charge for a while while main battery is out.
You're probably right. Rechargeable batteries typically last 5-7 years, in my experience.