OK - finally back to radio stuff - sorry for the delay.
Yes, Q3 is the 2N3906 and is installed per the silkscreen E/B/C holes. I re-soldered all three connections, just to be sure. Also, that R4 jumper reads just a tad under 3 volts when power is applied. Same voltage as at the base and collector of Q3.?
?Just taking a wild-ass guess here, but since the LED is lit, it is getting voltage and shouldn't be until the key is pressed OR if Q3 has given up the ghost. Yes?
?Perhaps replacing Q3 is worth a try - I'll see if there are any on Amazon that I can get without having to buy a bazillion of them.
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K7TXA / Eagle, ID