So I ordered a few MMBT3904 transistors. I removed the old one and soldered the new one on. I now get readings with the resistance check, but still just static (no sound). I didn't see your updates to this post until after I had already removed the old transistor. You mention testing it with a diode check is the only accurate way to do it, so now I'm doubting I was even testing it correctly to begin with... Most online tutorials have you test a transistor using resistance (the negative lead on the emitter or collector, positive on the base should give you numbers in the MOhms). I have a diode test mode on my DMM, but I'm not sure what I'm looking for. The numbers I get are in volts instead of ohms... As you can see, I'm confused.
Do you know the part number for the op-amp? The writing on it says "TL071B 804". There appear to be a few different TL071B chips out there.
Anyway, now that my original diagnosis is in question, is replacing the op-amp still the next best thing to try?
I'm not quite ready to give up yet, but just out of curiosity, what is the typical going rate for you or others to troubleshoot and fix boards?