To check to see if something is still 'on' when it is supposed to be off, use a multi-tester to check current draw accross the battery(when you get a new one). If there is current draw, even a small amount can run down a battery pretty quickly.
If there no current draw, then it is a bad battery. If there is, start checking the current draw accross each component. When you find the component that is drawing current, you probably have found your problem.
I'm not sure if the electronic dash uses any power to keep the memory for Odo, hours, clock, etc... but if so, it would be VERY small amount.
Once you get a new battery, if you don't find something using power, check your voltage after each ride. If you are getting lower and lower readings, it might be the 'alternator' (do these have alternators is the car sense - I haven't really looked!).
Good Luck.
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