I have had some issues with bad solder connections on the picture tube's heater circuits, located on the High Voltage transformer section. In most cases, if you live in a larger town/city, TV Techs will generally just do a job like that for $100 to $150 at most.
It's kind of a fast/easy fix for us. It gets much more complicated for the average Joe though.
That symptom you described where the picture gets darker after 5 minutes is due to the solder connections being cracked, and as the TV warms up they pull apart; your TV's heaters cool down you get less output from the picture tubes and once the connections cool down, they re-establish their connections and your picture gets brighter and the vicious cycle starts all over again.
If you have any more questions, just ask.
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