I have a TT600RE too, same year, 2003.
and last summer i had a very similar problem as yours.
no headlight, high beam lights on dash ON, etc... after a long time spent trying to understand all the possibilities, it turned out to be the bulb-holder that needed to be replaced.
as per the oil level increasing or decreasing when the bike is not used, it's normal.
the important is to make sure you don't have too much oil in the engine... especially after running it for 20 minutes.
good luck!
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