Sodium metal burning off when conducting electrolysis and I don't know why
First, sodium burns very nicely in air. However, it also burns in chlorine, returning back to the $\ce{NaCl}$ with which you started.

Most welders provide alternating current, meaning the the electrode becomes alternately positive and then negative (50 times per second [50 Hz] in much of the world, and 60 Hz in North America). Therefore, both $\ce{Cl2}$ gas and $\ce{Na}$ metal are produced at each electrode and recombining.

To produce sodium metal you need a direct current supply, such as from a battery, and some means to separate the chlorine bubbling out at the positive (anode) electrode from the sodium metal at the negative (cathode) electrode. A diagram of the Down's cell should help you understand this, and you can view a video of sodium production.

As a demonstration, I've made (for a very short time) small amounts of sodium in an open crucible using a laboratory 10 volt, 6 ampere supply... One sees a small metallic bead forming under the molten salt at one wire and immediately burning when it hits the air, with sodium's characteristic yellow glare.