If you press with your hand and feet down the center of the kayak (not on the sides), even the most unstable kayak is rock solid. If you can't bend around to do that, then you should see if you can use the paddle for support. You can put it behind the cockpit extended to one side and resting on shore, then hold on to the paddle as you enter. You weight will keep it in place, especially if the rear of the kayak has some sort of channel made specifically for this kind of situation so the paddle fits in it nicely.
Practice in warm shallow water -;)
Straddling is an option too, if the kayak is not of the 4-foot wide variety...
If you don't mind getting your bottom wet, you might consider a sit on top. All sorts out there, with makers like Epic Kayaks having high-end lightweight versions like the V6 touring that would also be easy to carry yet fast on the water.
Touring Kayak Paddles
Kayak & Canoe Covers
Sport Cases (Electronics)
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