You can train through this, but don’t miss this opportunity to use the information from your muscles to improve your technique as much as you can while your complaining arms give you the incentive.
In my experience kayaking does have a sort of breaking-in period during which a bunch of muscles that don’t typically get used in a specific way gain strength and gripe about it. After a few good trips you reach a plateau in which arms and hands are no longer the limiting factor, you really are doing the main work with your core, and your range in a day becomes more a function of general conditioning (/rest/nutrition on that day).
In addition to the excellent answer from Martin F above, I would emphasize the following:
—go out and paddle (with whatever form) until you get good and sore. Then at that point really focus on torso rotation and pushing the boat with your feet. It’s easier when you’re tired/sore because your core will not be as fatigued as your arms and will give you immediate pleasant feedback that you can move the boat while sparing your arms.
—in particular focus on a short power stroke that happens ahead of your hips. Most of the forward propelling force really comes from an amazingly short part of the time the blade is in the water pretty soon after you plant the blade. A lot of what you do with your arms is guiding it cleanly in and out.
—experiment with how loosely you can hold the paddle (especially with your top hand) until you’re on the verge of dropping it (you will instinctively grip tightly if the wind/water is actually in any danger of taking it from you).
—a drill I find useful is to make yourself paddle without bending your elbows. Seriously lock your arms straight out like an old horror-movie monster and figure out how to move the boat, even if it takes a couple of minutes to go anywhere at all. What you realize is that a big torso rotation and a very short very powerful stroke soon after you plant the paddle really does move you. Then add a little arm bending but as little as you can. Revert to this drill now and then as you find yourself drifting back into relying on your arms.
—try a bunch of paddles and buy your own even if you don’t own your own kayak. It makes an enormous difference. I like my hefty wooden Greenland paddle (carrying it on land it’s a heavy piece of lumber, in the water it’s buoyant and bears a lot of its own weight). Among European paddles, carbon fiber shafts and blades eliminate work spent holding up the paddle instead of moving the boat and will spoil you for ever wanting to paddle with rental gear. Bent shafts save some people’s wrists, wing paddles add undeniable speed and range (at the cost of versatility of control strokes). I personally refuse to feather my paddle (rotate the blades relative to one another), mostly because I’m used to a Greenland and don’t want to get confused on my brace strokes but I don’t accept that outside of a race the aerodynamic benefit outweighs the ergonomic drawback of adding even the small twisting motion. The point is experiment and find what you love.
—finally to answer your original pre-edited question, find anything you like to do that works your arms and hands whether it’s rock climbing or baking bread. It’s about lifetime habits there’s only so often most of us can force ourselves to train.