There are a lot of ways to get protective coatings that aren't plastic coatings. See if someone can do a flood UV coat. If book covers can be protected and not smell like plastic, then your cards can be protected and not smell like plastic.
If your vendor is a fairly cheap print-on-demand vendor, it could be that they're using the lowest quality processes to get your prints to you. It could also be the prototyping process itself. If your sample/prototype was free, it's likely it was printed digitally.
Will your final deck be printed in the exact fashion that your sample was printed? Depending on how expensive your deck will be to manufacture, it could be a good idea to pay a bit to get the exact same machinery running your full production order to also run a sample, so you know exactly what your end result will be.
Find other manufacturers, ask them what processes they would use to make your cards, and then ask how they would make a sample or prototype deck.
I'm making my own deck as well, and have contacted well over 20 vendors, both in the US and overseas, though my specific situation has a much higher budget than the average tarot deck. I'm still not 100% set on a vendor yet, but I've been talking to them since November/December. Getting the absolute perfect deck takes a lot of work and coordination.