Whenever lights do weird stuff like this the first thing I look for is a failed ground connection. You could start at one or more of the lamp sockets and measure resistance from the socket shell to battery ground, a clean bare spot on the frame/chassis, etc. Could also try testing it at night -- often when it's a bad ground, you'll find the lamps actually do light, but very dimly, and other lamps do too (indicating that the current is going through those lamps instead of directly to ground).
As far as the relation to the transmission selector -- maybe the neutral safety switch is involved somehow? Short of that I'm with frank-id; interaction between the transmission and turn signals would seem unusual at best.