no idea about panhandler, my guess is it would be a waste of money. You have gotten some good advice here but I'm not sure you heard it.
option 1 -
"No one will be able to give you a guaranteed fix to an unknown problem
They built on a wet land
This is what happens
Unless the land is HIGHER then some surrounding land then water will sit at the lowest point
Drains do not help unless there is somewhere for the water to drain TO
This is a good assessment of the situation - The solution is to re-grade and let gravity do its thing. It sounds like your mess of pipes are one bandaid already tried. You may need to incorporate a swale or stream. The general rule is 1/4" per foot of slope for run-off. In your case, it sounds more extreme, you may want to slope 1" per foot or more. A good engineer like ddman would be able to do a proper assessment and recommend a fix. No one will guarantee a fix.
Here's a suggestion: Let nature take its course to a certain extent. Take some of the areas that are mucky and that don't seem to dry-out and naturalize them. Instead of grass put in native plantings at these areas. There are native plantings, natural to your area, that are wet tolerant and drought tolerant. They have deeper root zones than grass and will help stabilize the soil and soak up the water.
This will be cheaper and may look nice but again, no guarantees. If you do just a little grading and create the "rain garden" natural wetland area you may be fine. I have also heard it suggested that weeping willows will absorb a lot of water. I do not know if hey will grow in your area.
Do this - First decide what outcome you want. Do you want hard surface like concrete? Then get multiple quotes for that. Do you want a lawn on the cheap? Go with the garden and some grading and hope for the best. Do you want to dry things out for good? Get a hydrological engineer and spend the time/money on a consult and implementation.
Deciding where you want to end up will help you focus your energies on the right solution for you.