Hi. No, most do not have batteries, and if they do, it's only for holding the station memory and your personal settings. The display should always work when plugged in. It sounds like you have a power supply fault. It is usually the capacitors drying up with age. You have had the first symptom - dimming over time. Now it is bad enough for the machine to mis-behave. Needs technical attention I'm afraid.
Mark, May 2008