Topography problem? Can you just create a drainage channel before the state classifies it as a 'wetland' and prevents you from draining? Or else fill it in ($$$).

Not sure what you mean about shifting soil - the water is washing it away? It is settling?

Otherwise roll with it and plant water-loving trees. I could recommend some, but I don't know where you live. Sycamore, cypress, silver maple, larch, cottonwood, - whatever is in local wetlands.

How old is the property?

If it is only a couple of years old, then the settling could be the natural compaction of recently graded fill, even if it appears to be in the trees.

Again if it is new, did the builder dig a hole for construction waste, or waste from the clearing of the property. If so it may be decaying and reducing its volume.

If either of these is the problem the only true solutions it to fill and bring the sunken area up to grade.

Easiest: sneak in a couple water tolerant trees. Like bald cypress, nyssa aquatica, whatever seems right for your zone.

More difficult: some/many(?) areas have rules regarding what you can do as far as changing drainage. For example, I can not build a mound to direct drainage off my property into my neighbor's. Check your regulations and fight it!

Some oaks even tolerate some water. I think Pin Oak is one. They might not reproduce naturally if submerged in the wrong seasons if I recall. Metasequoia is pretty damp tolerant if you are in zones 5, 6 or 7.... some red maples....not to the point of swimming though.

Lucky you! Did you know Pickerel Weed is in full bloom now and really beautiful. Introduce some Bald Cypress or Pond Cypress and work with the changing water table, don't fight it.
{{gwi:376415}}

"Perhaps only coincidentally, problems seemed to begin a few seasons after a neighbor cleared his wooded lot of trees."

It sounds like a drainage issue. If your neighbor cut down the trees on his lot, he may also have done some grading to make his lot more useful (to him). Even minor grading of the established drainage can have disastrous effect upstream.

Is there a possibility of a water, sewer, or storm drainage pipe going through the area. Those will leak from time to time and bogs to develop.