Well, the list is too long to get into. I will offer my recommendation, based on my own experience. A boat that I think would satisfy all of your criteria is a Current Designs Sirocco, or Gulfstream. They are the same boat; the Sirocco is the poly version and the Gulfstream is composite. Both are very well built and capable of just about anything.
I would suggest to you that if you are new to kayaking, you will probably think that nearly any sea kayak is going to feel tippy. That should soon go away with a bit of time in the cockpit. It can take a little time to gain confidence in any boat, but you will be amazed at how seaworthy these wonderful boats are (sea kayaks).
Keep an open mind about the boats you try out and do not be too quick to judge their feel and performance. Some of the things about a boat that might seem all wrong the first time in it, might later be the very aspects that you will come to appreciate. Take the time to learn what constitutes a well built boat and which construction material suits you. Spend some time reading reviews of boats on paddling net., but understand that you will get many opposing views.
After you've got a few hundred, or thousands of miles and hours in the saddle, you will probably be interested in upgrading. It starts all over. I have some very specific opinions in that regard, which I am happy to share.