Short Keel vs Normal
You seem to be asking a couple questions here and your terminology leaves me a little unclear as to what you are asking so this may be a more comprehensive answer than you were looking for.
I assume that when you are using the terms shorter or longer you are not refering to the depth of the keel but its length along the hull. These days the more normal keel is pretty short and is called a fin keel. More on that below. By stability, I am assuming you mean the ability of the boat to stand up to its sails without heeling, rather than directional stability, which is the tendancy for the boat to go straight. The fore and aft length of the does not affect the stability.
Stability is controled by the shape of the hull, the vertical center of gravity and to a lesser extent weight. The lower the center of gravity the better the stability and the more comfortable the motion. Fin keels tend to be deeper and therefore generally have a lower vertical center of gravity. Here is a synopsis of the various keel types:
These earliest keels pretty much ran from the point of entry at the bow, to the aft most point of exit at the stern. Those are full keeps in the fullest sense of the word.
They have some advantages; they theoretically form a long straight plane which keeps a boat on course better (greater directional or longitudinal stability). If you run aground they spread out the load over a larger area reducing the likelihood of damage. Once really planted aground they reduce the likelihood of the boat from tipping over fore and aft. They are easier to haul out of the water to work on. You can spread out the ballast over a longer distance and so they can be shallower. You have a greater length to bolt on ballast so it is a theoretically sturdier and simpler connection.
They have some disadvantages; they operate near the surface and near the intersection of the hull and keel which are both turbulent zones so they are less efficient in regards to reducing leeway for a given amount of drag. They also have comparatively small leading edges, and the leading edge is the primary generator of lift preventing sideslip. Because of that they need a lot more surface area to generate the same lift. Surface area equates to drag so they need more sail area to achieve the same speed.
As a boat makes leeway water slips off of the high-pressure side of the keel to the low-pressure side of the keel and creates a turbulent swirl know as a tip vortex. This is drawn behind the boat creating drag in a number of ways. The longer the keel bottom, the bigger the vortex, the greater the drag. So they need more sail area again to overcome this drag. To stand up to this greater sail area the boat needs more ballast and a stronger structure, which is why long keelboats are often heavier, as well. (Of course more sail area is need to overcome that weight as well. It becomes the classic weight adding weight spiral. It is important to remember that weight, in and of itself, does nothing good for a boat. It does not make it more stable, comfortable, able to carry more weight, or stronger.) Long or full keels tend to be less maneuverable.
In my book, any keel that is less than 50% of the length of the boat on the bottom of the keel is a fin keel. Fin keels came into being in an effort to reduce drag. Cut away the forefoot or rake the stem, as well as, move the rudderpost forward and rake it aft and pretty soon you have a fin keel. Today we assume that a fin keel mean a separated rudder (skeg hung or spade) but in fact early fin keels had the rudder attached in a worst of all worlds situation. A lot of classic CCA rule beaters are actually fin keel with attached rudders which you often see misadvertised as full keels. They are not.
Fin keels with separate rudders seem to be the most commonly produced keel form in the US these days. (I could be wrong, there is a resurgence of full keels these days)
Fin keels have some advantages as well. They have less drag as explained above so they typically make less leeway and go faster. You can get the ballast down lower so in theory they are more stabile for their weight. They are more maneuverable. They take better advantage of the high efficiency of modern sail plans and materials.
They have some disadvantages as well, many of these have been offset or worked around by modern technology but at some level they are still accurate critiques. They have less directional stability than long keel boats so the tend to wander more under sail. Since directional stability is also a product of the dynamic balance between the sail plan and underbody, in practice they may actually hold a course as well as a full keel. In general though you can expect to make more course adjustments with a fin keel. It is sometimes argued that it takes less energy and is therefore less tiring or the autopilot uses less battery capacity to make these corrections so a fin keel may also require less energy to maintain course. This I think is a product of the individual boat and could lead to a debate harder to prove than the number of angels that can dance on the head of a pin.
Fin keels are harder to engineer to withstand a hard grounding and when aground they are more likely to flop over on their bow or stern. (Although in 42 years of sailing, I have never heard of anyone actually experiencing this.) Fins typically have deeper draft. They are easier to pivot around and get off on a simple grounding.
A shoal keel is just a keel that is not as deep as a deep keel. Today the term seems to be applied mostly to shallow fin keels. Shallow full keels seem to be referred to as shoal draft boats. A shallow fin is a tough animal to classify. I really think it has few of the advantages of either a deep fin or a full keel and has many of the worst traits of both full and fin. This can be partially offset by combining a shallow fin with a centerboard, which is a neat set up for shoal draft cruising.
A lot can be done to improve a fin. One way is to add a bulb at the bottom. A bulb is a cast metal ballast attachment added to the bottom of the keel. They concentrate the ballast lower providing greater stability and sail carrying ability than a simple shallow keel. Traditionally bulbs were torpedo or teardrop shaped. They have been re-contoured to provide some hydrodynamic properties. Remember tip vortex from above. Shallow keels need to be longer horizontally than a deeper fin in order to get enough area to prevent leeway. This means that it would generate more tip vortex and more drag than a deeper keel. The bulb creates a surface to turn the water aft and prevent it from slipping over the tip of the keel thereby reducing tip vortex. This does not come free since a bulb increases frontal area and surface area. A Scheel keel is a very specialized bulb that is shaped to reduce leeway and drad.
Wing keels are a specialized type of bulb keel. Instead of a torpedo shaped bulb there are small lead wings more or less perpendicular to the keel. These concentrate weight lower like a bulb and properly designed they also are very efficient in reducing tip vortex. There has been some discussion that wings increase the effective span of the keel when heeled over but this does not seem to be born out in tank testing of the short wings currently being used in production sail boats. Not all wings are created equal. They potentially offer a lot of advantages, but their effectiveness is heavily dependent on the quality of the design and I really think that many wing designs are not really working to their potential.
I am not sure that this is answering your specific question or not.