Sheild not needed
What fmarino1976 said. Some people do use shield, but its not necessary. I don't know about the noise you hear just as you key the mic. What kind of radio do you have? Is it an old radio?
The PTT on modern radios is a ground signal, separate from the radio receiver/transmitter and audio amp circuits. The PTT ground is isolated through a transistor or electronic switch.
fmarino1976 is probably right about the switch being dirty or poor quality. If its making an intermediate contact, it could telegraphic into the radio, possibly, but I'm not 100% about that. The switch would have to be real lousy or dirty. It's easy to check, by putting in a new, better quality PTT switch.
If a new PTT does not do it, it could be something else on the headset audio side of the system and not the PTT side. I agree shielding on the wire is not needed.