There is really no such thing as a “beginner kayak”. Get the best boat suited for your purpose and you will learn to use it. If you are learning to drive on a highway you don’t start with a golf cart and if you plan to ride a bicycle on trails you don’t start with a single speed bike with training wheels.
10’ boats are a poor choice for open water such as you are describing - they are for shallow creeks and small ponds. 12’ is a bare minimum and for any place where you could encounter off shore winds, currents and waves you would be more secure in a 14’ or longer boat. The question isn’t stability, it’s whether you can get back in to shore in rough weather without exhausting yourself. And whether the boat has enough displacement to allow you to climb back in easily should you dump it.
I disagree with Dave on Academy Sports as a good choice for your first boat. I have been unimpressed with their lines of boats (heavy and clunky or flimsy and cheap models for the most part) and the lack of kayak-specific knowledge of their personnel. Better to find an outfitter who specializes more in wilderness sports like kayak touring and backpacking. I have yet to see a store that sells golf clubs, hunting gear and soccer jerseys that can responsibly advise a beginning kayaker on appropriate purchases.
With a budget of $800 you should be able to find a decent used boat, even with a good paddle and PFD thrown in. Folks starting out are usually better served by buying used - have some patience and keep looking at ads on Craigslist – don’t forget the classifieds on Paddling.com too. You can search by your local area. This is the time of year when people start cleaning out their garages for winter and some good deals can be had.
Also, outfits that rent and sell kayaks often liquidate their rental fleets and demo boats before the end of the year so that they can take a tax break on them (otherwise they may have to pay tax on them as “inventory”). If you can find a good dealer in your area, ask them if they have any boats like that or even trade-ins.