So first off, if you're like me you want to do everything right away. And you can justify it because those trees you want will take most of a decade to start producing so they need to get planted NOW!
You'll wear yourself out. Do a little bit at a time and you'll last a lot longer. We just bought our first house (on an urban lot) and between fixing it up and getting ready to plant a forest garden I have my time accounted for at least through most of the coming year.
Chinampas come to mind, though it doesn't sound like you have enough year-round water for them to work. Basically you dig channels and build raised areas in between. I think it must be a tactic for utilizing swamps. Anyhow you can do
aquaculture in the channels and the raised areas are watered but not waterlogged.
All this is kind of foreign to me since I'm in CO and needing to deal with too much water has never been a problem.
Your Hugelkulture idea sounds like you might be on the right track. My brother built one and was disappointed (Colorado, so rather dry and the thing never really composted. I think for it to work right you need lots of green stuff in with the wood. And since it's composting, it might take some time for it to be ready.
It seems like increasing water infiltration will be key. And figuring out what to do about the runoff from your neighbor. It could be an asset or a liability.
Deep-rooted cover crops would be a minimal-effort strategy that might loosen up the soil. A subsoiler might be a good option at some point, too.
A strategy I'm using is deep wood chip mulch, 12 inches is the target, to maintain moisture and build soil. In your case it'll decompose a lot faster but I wonder if as it decomposes and the worms work on it it would improve your infiltration rate. Silt/clay sounds like it would get waterlogged easily. I've gotten a couple loads of wood chips from the local tree trimming company (for free). Paul in the back to Eden documentary composts the wood chips and uses them for his garden, so they're multi-purpose.
Most of all enjoy your new place!
Daniel