I had to draw a small diagram to see how the area was around the house. I drew a swale in to show what is there already, or hopefully something that can be created
![enter image description here](https://i.stack.imgur.com/2XwEi.png)
What looks troubling here is the concentration of runoff heading to the house IF there is no swale there, that is why I hope there is one there, if not you really need to get the flow away form the house. It is bad enough that it is sandy, the water can still perk into the foundation, though it will be minimized with a swale. The clay base on the north side, sounds exactly as you describe, a bowl.
You need to do whatever you can to minimize the water infiltration, to start. It may not require to pipe the downspouts away from the house to a drywell or into a French drain, well not a French drain as you may plan around the house. French drains return the water to the ground, and in turn the crawlspace. What I mean is terminating the downspouts into an in ground pipe that daylights downhill of the home. A drywell would work too, but it is much more than may be needed.
This is the minimum I would suggest, although it is still a lot to do. I would also consider waterproofing the foundation, it may work on most of the foundation, but I think the runoff would still migrate under the foundation on the south side since it is still on sand/ clay mix presumably.
All of this is still tempered by the symptoms created by the water in the crawlspace. If there is no evidence of black marks (mildew) or white fuzzy stuff growing in places from the excess moisture in the crawlspace, if none of these symptoms exist, al least extreme measures are not needed. Regrade if needed, would be the minimum, downspouts to daylight away from the house the next active remedy, drywell addition the next, and if you are doing all that digging, waterproof the foundation, all these measures combined are the extreme I mean.
![enter image description here](https://i.stack.imgur.com/ELVDq.png)
Edit 1/26/13
Aside from the shape of the house and other extensions from it, it shows the same thing happening except your swale that exists is more of a collection point. It appears the rain runoff takes the same path, but the house is in the way. The swale in the above sketch I posted would divert the rain water before it gets too close to your home. Is this a possibility to happen? It can be done by lowering the grade or if the house is high enough above grade, the grade can be raised, but that has implications too. The water in the swale may be of no concern. I would draw more concern to the south and west side where the rain looks like it is path is unencumbered to the foundation.
I will edit your drawing to show a more accurate idea and post it.
Back at the downspouts that I mentioned earlier, if the grade looks like it can daylight a underground lead would help reduce the amount of water at the foundation. Yet the addition of the swale will redirect the downspout water as well.
It would depend on soil type and the load on the foundation. As a very rough rule of thumb, the soil actively supporting a foundation spreads at a 45 degree angle downwards from the outer lower edges of the foundation. You should be able to dig outside this section with impunity. If you encroach upon it, if you only excavate something like 15 foot sections at a time, assuming the wall is longer than this, you shouldn't encounter any issues unless the foundation is extremely heavily loaded, such as supporting a large concentrated load.
When backfilling, be sure the soil within the support prism is very well compacted, but be careful not to collapse the pipe.
Best Answer
Look at a typical "dry well" install - basically terminate the pipe into a blob of crushed rock, or an actual void/hole/tube punched full of holes, so the water can filter out. Just stopping the pipe underground will not work well.
In an essentially non-freezing climate there should be little problem with this - they can be done well in freezing climates, but it's more work and digging. If you use a open/void/hollow well it's easier to check/clear the pipe if that's ever needed.