Just a not-often-discussed topic since you're a newbie to germinating palm seeds...there are three main types of palm seeds found in the tropics and subtropics: orthodox, recalcitrant, and intermediate.
Unlike temperate-zone and desert seeds, which usually can be stored dry and even frozen, only orthodox seeds can handle long periods of dryness or cold temperatures and still remain viable. Recalcitrant seeds are sensitive to a lack of warmth and moisture, and will fail if exposed to "standard" dry-seed-storage practices. Intermediate types can handle a longer amount of dryness and usually some amount of cold (probably not really freezing in most cases) and can be stored a little more loosely for a longer period.
You can look up the status of a species' seed with a google search. But you can also guess if you know their native habitat. Rainforest palms, since they usually drop their seeds into a place with constant warmth and moisture, will likely be recalcitrant; palms that grow in tropical/subtropical areas with a marked dry season and monsoon-style moisture (like Western Mexico) may be intermediate; and those from drought-ridden or desert areas will likely be orthodox, for obvious reasons.
Since the seed-dealers you find online (Etsy, eBay, et al.) generally have no idea about this, it's best to either buy from a known palm-savvy grower (as here on PalmTalk) or collect your own fresh seed and make sure, as Meg mentions above, that they have not been devoured yet by borers (this is very common in fallen seed in Florida). Best is to watch for the "season" for a species and then grab the seeds off of a tree before they hit the ground. Clean off the fruit by soaking/fermenting (you can place in a water-filled baggie and massage daily and keep rinsing until they're clean) and then put into a clean baggie with just-moist (not wet) peat-moss/coco-coir/perlite/sand or a mix thereof; or in a greenhouse in either community or individual pots. Then wait. The two species you have are both very, very slow. Thrinax faster than Coccothrinax. they will look like blades of grass for two or three years before they start to develop thicker leaves and even longer before you see their "character leaves."
Since blades of grass are going to be very depressing for any newbie, you might also get some Veitchia (Montgomery Palm) seeds or Adonidia (Christmas Palm) seeds. Also a coconut or two. They sprout quickly, throw up large leaves from the start and will get you very excited, particularly the coconut and the Veitchia, because they grow like rocketships. And then you might want to go down to Homestead when you have a chance, find a nice inexpensive mom-and-pop nursery on/around Krome Ave. and buy some Thrinax or Coccothrinax palms (the fastest Coccothrinax is C. barbadensis, often sold in SoFla as "Coccothrinax alta") to enjoy while you spend years watching your seedlings grow into something. But be careful, because, slow or not, seed-sprouting can become addictive!