Did you find clay at the dry area after digging down 1"? Clay can prevent water from permeating the ground. You may want to check the soil further down the slope at the dry area to determine if there’s clay there. If not, that soil will percolate allowing it to dry.
You can rent soil aerators. Look for a model with hollow spikes, so you remove a plug of soil. Solid spike models push the plug, compacting the soil further in that little spot.
If you do rent an aerator, as purple suggests, and I’m not disagreeing, here is my advice based on one season spent aerating lawns for true green many years ago;
Aerators are (or can be) large, heavy, clunky cantankerous machines. True Green rents, or rented, all of the ones they used. There were a LOT of machines that had pretty badly worn key components. Such as drive sprockets for the chain drive that moved the spike wheel and the drive wheels (yes they should be self propelled, remember, heavy) so we always had to have a wrench and a screw driver on hand to put the chain back on and readjust the tension. Sometimes when the chain came off it would slip between the drive sprocket and the frame becoming seriously wedged.
So the advice part, make sure the rental agency delivers and picks up, if you can even get one and that they aren’t going to hit you with some bs repair bill for when the chain inevitably slips off the very worn and poorly maintained machine
And yes you want a plug puller not a plug pusher