Your old one-wire signals did indeed ground through the "stalk" - your new one have running lights built in, thus two filaments.
Your 350 is not wired for the new ones.........if there are 3 wires, one is ground, one is running light, the other is the turn signal.
You can rig it up to work as a turn signal only, just figure out which wire is which. Run the ground to any available green wire on your 350, the turn signal wire to orange or light blue, ignore the 3'rd......