Vermouth for use in Drinks - can I freeze it?

Water freezes at 0° C (or 32° F), but ethanol freezes at -114° C, so you can guess your vermouth with freeze somewhere in between those two temperatures.

Noilly Prat is 18% ABV, or 15% alcohol by weight, which means it would freeze at around -6° C to -7° C, or 19-21° F. Your freezer may or may not be tuned to be that cold, but if not, you should be able to crank it a bit and get your Noilly Prat just frozen.

That said, I would not really recommend leaving an ice cube tray full of vermouth hanging out in your freezer. You may notice that old ice tastes funny. Even though you may not notice your freezer smelling, that's partly because the cold blocks smells. Just because you don't notice them doesn't mean the odors aren't there to ruin your Noilly Prat. I would recommend leaving them for a day until just frozen (covered if you like), and then moving them to a Ziploc for long term storage.

All that said, freezing will not halt oxidation altogether, but it will help, as only the exposed surfaces areas should be able to be oxidized. Since ice cubes by design have a relatively high surface area, you may consider minimizing the exposure to oxygen by other means.

If you have a vacuum sealer, using that would be a fairly simple means to limit that exposure, but a similar effect can be done with a simple Ziploc bag, as described here (note this link is focused on using this technique for cooking, but it is translatable to your scenario). Drop your ice cubes in a bag, and then slowly lower the bag into a container of cold water (the colder the better, here, to minimize melting), leaving the seal exposed. I sometimes clip things onto the bag to weigh it down as well. Then begin closing the seal, and lower the bag further into the water as you continue sealing, so that only the unsealed portion is exposed. Finally complete the seal and you should have a reasonably low-oxygen environment to mitigate further degradation of your vermouth.

There is no easy answer to this problem. The consensus seems to be to try and buy small bottles where available (difficult) and always refrigerate. But throwing away after 2-3 weeks seems a bit extreme. Vermouth doesn't tend to go off completely straight away, it just gradually loses character. Its also worth noting that Sweet Vermouth tends to fair better on the shelf than Dry.

Vermouth is white or red wine that has been infused with a mixture of botanicals and fortified by the addition of a neutral alcohol like un-aged brandy or grain alcohol. The fact that it is fortified leads many people to believe that it is shelf-stable, that is simply not true. For a better vermouth experience, buy a high quality product such as one of the offerings from Boissiere, Noilly Prat, or Vya, and always buy from a source with high turnover. Vermouth should be used within 6-8 months of bottling or it begins to go off. Once opened, it should be stored in the refrigerator and away from light. Even when stored properly, it oxidizes like any other wine, so it is advisable to finish the bottle within a month after opening. Unfortunately, this means you should almost never order vermouth (or a cocktail containing it) in a random bar where the bottle is just sitting out and has been open for who knows how long. Find a cocktail establishment where they care about these things, or make it yourself at home.