The fact that the screen illuminates properly and does not stay absolutely pitch black means that the backlights are working properly.
A faulty LCD Controller Board (or TCon board for short) could possibly cause the symptoms you describe. A replacement board can be purchased on Ebay for around $60 plus shipping. See the following link for one example:
http://cgi.ebay.com/SONY-KDL-32S3000-T315XW02-T260XW02-CONTROLLER-BOARD-/220731395519?pt=LH_DefaultDomain_0&hash=item33649dcdbf
However, if you are unable / unwilling to disassemble the unit and replace the component yourself, then you will seriously have to consider the cost benefit of risking more money trying to repair the unit. Sony products are generally pretty reliable and worth the effort of repairing, but a technician in the repair shop will probably charge you at least $100 just for the labour to replace the part, plus the cost of the part itself. I don't know how comfortable you are with repairing electronics since you said you had the power supply board replaced by someone else. You will have to weigh this cost against the price of a brand new TV which comes complete with a warranty.
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Thank you very much and good luck.
Joe.