i too have a very high water table that runs down off the hill top I live on , it's just a few inches below the surface most of the time. every time I dig a hole the glacial clays soon see it filled and stay filled for days at a time.
in addition to brown eggs suggestions .
if you have any chance of limestone dust from a quarry get it and hand braodcast it over the areas in winter and spring but not at the same time as manuring for the acid manure will kill the limestone and vice a versa .
do them at least 20 day a part then a month later do a soil test to see where you are wrt the pH ( parts of Hydrogen ) of the soil and adjust if needed to get a pH for what you want to grow. .
There is a very old way of draining things quite a bit a bit if you have some gradient to drain through the easiest if you can affort to hire a mini digger and get it to where you want it is to dig a drawn to scale set of herring bone trenches some three feet six deep and fill the bottoms with laid straw and brush wood faggots to a depth of 15 inches or so or again if you can afford it use perforated land drain wrapped in the soft black weed membrane material to help keep it clean then lay it ona 12 bed of 10 mm clean crushed stone and cover it with the same amount then back fill .
There is a modern day equalivalent which uses plaited straw some four inches thick for the water route instead of a a plastic pipe.
The straw and faggots should be Ok for 15 or more years as there will be little oxygen or microbial action to decay things that deep down .
You can do this by hand but it really is hard graft and takes ages.... I know I've done over two hundred mtrs of trench else where for the same reasons.
Whilst doing this set of gardens I've got hold of several small two tonne truck loads of bark chippings & coarse saw dust from a saw mill and used that in with the back fill over the land drains so I get some water percolationg down through the stuff into the drain pipes. I've also mixed /.spread rotavated in several 1 tonne bags of sharp builders sand into the top 10 inches or so of the garden and lawn areas.
I've done this to at least give the lawns a fighting chance of survival instead of them ending up like most of my neighbours areas of soggy cut moss.
Spiking your lawns / grassed areas and hard garden then broadcasting sharp sand mixed with decent compost and a bit of dried old grow bag contents over the holesthen stiff brushuing it over the spiled area is also beneficial for areas where you walk on, as not only does it give a sort of drainage it also oxygenates the ground and helps the worms get going , I've usually hired a petrol driven machine as it much less effort .
My mates laughed at me when I spiked the large cut grass areas at our old place , they laughed even more when I dragges an eight foot long 6 foot wide bit of weighted down chain link fencine behing the ride on mower over them in early spring but boy were they envious of the lawns in summer.