The pieces you describe are indeed the main wear parts in the turn signal switch. They're called the shuttles, and when they're worn out you really do need a new switch assembly. No soldering is necessary; a new turn signal switch comes with the wire harness already attached and all the terminals already installed on the ends of the wires. You'll be replacing the switch, wires and all, in one go.
The turn signal switch wires connect to the underdash wires with a snap-together nylon connector block. Separate its two halves, then remove the nylon connector block by releasing each individual wire from the block, one at a time. This is achieved by means of a terminal extractor pick (or, if you're in a hurry and don't want to spend money, a darning needle or unbent sturdy paper clip or mini screwdriver), inserted into the slot at the "front" (connector) end to depress the terminal's lock tab. Then a careful tug of the wire pulls it out the back of the connector block.
Draw yourself a diagram of which wire goes to which slot in the connector block, especially if you don't have a wiring diagram for the car (you can download one free of charge
here).
Once you've released all the terminals from the block, tie a long piece of string or twine round the bundle of wires at the very end near the terminals. Withdraw the old switch, which will pull the string up the column. Untie it, tie it round the wire bundle on the new switch, then pull the string to guide the new wire bundle down the column without any of the wires snagging and refusing to show themselves at the bottom. Then just snap each wire into the nylon block in its correct position, plug the two halves of the connector together, reassemble the switch and lever, reinstall the steering wheel, and you're all set.
As for where to buy the switch: They're around; not impossible to get, but shop carefully. Interchange on that switch is '64-'66. The vendor with the name starting with "L" mentioned by another poster in this thread has reputation problems; see threads
here,
here,
here, and
here, for just a few of many examples. :-(
