I'm not familiar with Eastman's, but have used another brand in the past (I like to mix my own cure & seasoning). You're best to follow the instructions with the kit, though. If it's just one packet of each (cure & seasoning), it's for a set amount of meat, probably 1lb, so be be sure you have enough before you start to prep for curing. Weighing your meat after trimming and slicing is always best, so you know how much you're working with and can adjust the amount of cure you mix for that batch of meat. If you have Morton's Tender Quick or Cure #1 you can make the cure amount as needed, and then of course the seasoning mix.
I will say this: if you mix the seasoning/cure packet with water (mixes I've used recommend this anyway...about 3oz per 1lb mix) it it makes it far easier to achieve a full application by tossing the meat in a bowl after adding the mix, then transferring to a zippered baggy. If applied dry, you'd need to work the meat a lot more to achieve full and even application. For 1/4" thick slices, allow at least 24 hours to cure...36-48 should result in a full cure while 24 hours may or may not (1/8" will cure faster)...depending on fridge temps. 36-38* is a good temp range for curing...too cold slows down the cure process...too warm (over 40*), and, well, it could obviously spoil before it cures.
Also, toss and squish the bag around a few times during curing to allow for movement of the meat and cure/seasoning mix for better assurance of full and even application, as when meat is touching together it can tend to force the seasoning/cure out from between the pieces...tossing it around helps to redistribute it throughout the batch on all surfaces.
I've never rinsed off the meat before smoking/drying, as the seasoning will stick to it pretty well by the time it's cured. Use what you know you like up front during the cure and all be be good. For soy, worsty or other sauces, they add salt, so it will be a bit saltier than you may like, although the sauce could replace part or all of the water for the mix. The additional salt from the sauces can also help for longer storage times, if you can make it last long enough to benefit from that...we never can, unless I stash some in my Q-Freezer under lock & key...LOL!!!
Hope that gives you some more insight.