When you said you were "too cheap to spend money on taking it to the shop", I took what you said literally. However, I get the sense you are more than willing to spend a bit of money and time on the TV, knowing full well you may spend more on unnecessary parts without the safety net of a certified repairman. Additionally, I gather it is more of a matter of principle or perhaps you want to take it on as a hobby or a sense of accomplishment. If that is what you really want to do, then you are not "cheap", because short of a couple straightforward general maintenance repairs, trying to repair this set through trial-and-error will not be "cheap".
The couple of maintenance issues to which I was referring are with regard to the lamp and ballast. A good ballast will have trouble firing an aging lamp. Once the lamp is lit, the picture may very well look good. By contrast a failing lamp will have trouble firing up even a new lamp. Additionally, a failing ballast will shorten the lifespan of a good lamp. If the lamp fails to fire altogether, the TV will attempt 3 times to restart. At this point, I'm not sure if you would still get sound without a picture as you are experiencing. Temporary sound, maybe, but I don't believe the TV will continue to run when I doesn't detect the lamp having fired---unless it is a video board failure. So, if you are getting sound with no video for an indefinite amount of time, I would be highly suspicious of a failing video board.
Otherwise, I would do the following:
- Reset the TV: Unplug it--> Hold down power button for 30 sec. -->Plug it back in.
- Replace the lamp with a new OEM lamp.
- Ensure the blue safety switch on the lamp door is secure such that it is in the down position when the door is closed. I would just tape it down to make sure.
- If still no joy, I would replace the ballast. (a simple procedure and the ballast can probably be had for less than $80).
After doing the aforementioned, I would spend no more money on this TV without taking it to a repair shop. At this point, you've essentially done some general maintenance on parts that are likely reaching their end of life anyway.