Explanation for a basic decomposition of water experiment
If the hydrogen and oxygen molecules take up the same amount of space, then why did the tube over the negative terminal have much more than twice the gas as the other one?


Good questions but keep in mind that you did not mention the concentration of sodium chloride. If you have too much salt in water, chlorine gas is produced (eventually bleach) instead of oxygen. This is why you did not see the expected 1:2 vol. ratios.
If possible, use the lowest possible salt concentration for the experiment or even better, can you use a small amount of Epsom salt? Then there is no issue of chlorine formation and only oxygen gas will form. You will certainly see 1:2 vol. ratio.


The atomic mass of oxygen is nearly 16 times that of hydrogen, so then even if there was twice as much hydrogen in the product, wouldn't the oxygen still take up more space?


This issue was tackled by scientists long time ago. The key point is that if you have 1 billion molecules of $\ce{H2},$ and 1 billion molecules of $\ce{O2},$ they will occupy the same volume at a given temperature and pressure. There is a named law behind it. The volume of molecules way too small as compared to the volume they occupy.
The radius of oxygen molecule is 152 picometer (symbol: $\pu{pm}$) and that of hydrogen is 120 picometer $(\pu{1 pm} = \pu{10^{-12} m}),$ so masses and volumes do not correlate exactly.