"so the land IS graded downward into a small swail..." It sure doesn't look like it. Your yard is obviously not PROPERLY graded. The pictures do not correlate the swale with the problem area so it's not going to be possible to suggest a specific fix. But I could pretty much bet that a french drain will not help you at all. A conventional collector drain connected to the swale by an underground pipe might be a possibility but one would need to have specific grade (elevation) information. Could you pull up a Google Earthview photo of the property to show how the problem area relates to the swale? (And post directly into the thread.)

Thanks for the reply. When I said that it is graded, I just meant that you can visually see that the ground slopes from one side of the property to the other, obviously not very well graded haha. I am attaching two pictures, sorry there are a bunch of trees so it is not that easy to see but the red dashes are the ditches or swales. Thanks for all your help!

IF it is true that grading across the entire lot, obvious to the eye, is sloped toward the swales, then you could fill only the depressions to the same plane as surrounding area so they can no longer hold water. Surface water could not pond if there were no depressions. It's interesting that you have an area of ponding exactly adjacent to the swale extending into back yards. There is a lip of some sort at the edge of swale? ... Or there is another depression concurrent with the swale? Specifics as to cause of problems are not really showing in the pictures so far.

So I took a panoramic picture, I don't know if that will help at all or if you can see anything but one image is the original and the bottom image shows my thoughts for drainage. My thought is to have a drain basin (shown by the red box) in two areas and connect them using the drain pipes with holes in it along the way to trap additional water and exit to the swale leading to the fence. A lot of the standing water by the backyard swale is because leaves and dirt keep getting caught in the fence and therefore the water does not get through. Thanks for the help.

"A lot of the standing water ... is because leaves and dirt keep getting caught in the fence..."

For sure, I would solve this problem first. There should be a gap below the fence that is large enough not to trap leaves. If one expressly wishes for there not to be a gap, then one should commit to a leaf removal patrol.

To note ... in addition to filling any depressions, if there are any humps or rises in the grade which prevent the movement of water, those must be removed. It's best done by a skid steer loader or similar equipment. I know people don't like to mess up existing lawns, but it's a bullet better faced than ignored.

I think your drain plan is likely to end up being a pain with ongoing clog removal and general insufficiency. I can't see how it's going to address the R. side ponding. Also, it's not as though there is a central area where water ponds that will be a good drain location. Holes in the pipe are just as likely to become clogged with tree roots and silt. I would vote for a drain if you had a single area that just couldn't be fixed (too expensive) with grading. But it's not the case. You have a lot of areas to fix and the long term trouble free solution is proper grading and not having any pipes and drains to clog.