At your height and weight you can probably be pretty comfortable in a 10' yak. Where you will see a real advantage to the longer kayak is in tracking when you are making a long paddle. The longer the kayak, the easier it is to paddle straight. Stability comes in large part from the width of the kayak. But the wider it is while being short, the greater tendency it will have to turn while paddling, especially paddling against current or wind. Naturally a wider kayak is also less hydrodynamic, meaning you have to paddle harder to get where you want to go. On small bodies of water that isn't a real big deal.
In your original post, you said you would be fishing smaller bodies of water. I think what your definition of small, how long you will use it for at a time, and how often you will use it will determine your length. If smaller bodies of water means bass fishing 5-10 acre lakes or floating down a small river to a take out vehicle 5 times a year, then you will probably be happy with a relatively wide 10' kayak. You can save some money and storage space, and still get out and have some fun. If you are hitting larger reservoirs or paddling up a river to float/fish down river, and doing so for a full day at a time, and doing so every weekend then the benefits of a 12' would be noticed. As one1bravo said, if you get serious you will want at least a 12'. Trailer hitch bed extenders are not too expensive.
If your lady likes to fish, but only occasionally, then buying a 10' boat to get your feet wet, and figure out what you really want out of a boat isn't a bad idea. In a year you could buy the bigger boat you want, and have the 10' for her to go out with you.
You don't need to go too crazy for a paddle. I have an Aquabond I bought on sale for $75 (5 years ago). I am sure there are competing brands that will also serve your purpose and you may be able to get for cheaper. When you start to get much above the $100 price point you start paying a lot to get a reduction in paddle weight. If you are paddling for long lengths of time, then the weight makes a difference. If you are mostly fishing, and only doing so for 3-4 hours at a time, then the more expensive paddles are hard to justify. My fishing buddy upgraded to a Hobie, and I used his $400-$500 Werner a few times. It was definitely nicer, but for what I do, I cannot justify that extra coin. On the subject of paddles, a paddle that is slightly too long is generally better than a paddle that is slightly too short.
As one1bravo said, the Hobies are where it is at for fishing from a kayak. It is amazing how much faster they are with the foot pedal, plus you can move while you fish. They are also very stable. There is no way I am spending that money at this point in my life though. The Hobie kayaks are also heavy as sin.
The nice thing about an accessory rail is that you can move the accessory around if a fishing buddy, in-law or ladyfriend hops in to take it out for a spin. I also move my pole holder position a fair bit when I change fishing styles. If I am trolling a lure I like my pole holders in a different spot than if I am anchored up bait fishing. When you mount front pole holders or something of the like they are semi-permanent in their placement. If I destroyed my kayak tomorrow, the two hard requirements I would have would be an accessory rail, and a lawn chair/raised type seat with ample back support.
You will also probably want to install an anchor trolly at some point. Even if you aren't fishing in current, the anchor is nice for keeping wind from blowing you around on small lakes. The anchor trolly makes it so that you can move the anchor point up and down the boat to position it where you want it and for easily retrieving the anchor. It is also a good diy project.
Best of luck on getting what you want.
