I would second the comments recommending a second opinion. A sump is in the basement to allow water to be drained away if it accumulates around the footings. If there's no water there it quite probably means that your water table is low enough that there's not seepage. It also would indicate that you have decent drainage around the house and rain water is running away and not down along the foundation walls.
Even if there is a blockage in your sump I would expect there to be some water in the bottom if you currently have a water problem. Blockages often involve crushed weeping tile or other drainage pipes. However, it would not usually be a complete blockage and if you have a have water on the outside trying to get in at least some of it should be seeping into your sump even through damaged draining lines.
If it's ground water, the issue isn't water vapor, but rather just plain water. Is the water table close or higher than your basement slab? If so, not a whole lot you can do to completely stop moisture issues, as that basement was just built in a bad spot.
That said, the main solution would be a sump pump and drainage tile system. Any water coming up through the earth dumps into the drainage tile, into the pump well, and gets pumped out. Again, though, if the water table is already higher than the basement slab, that'll be a never ending battle as well.
That said, perhaps the issue isn't as much about ground water as you think. Does it get humid in your region? If so, then 'damp' basements are par for the course. Basement walls are usually always going to be cooler than the air, and, a such, will be were condensation forms in a humid environment.
To prevent that, you need to a) dehumidify and/or b) insulate the walls.
A dehumidifier constantly running tends to be a normal part of any basement in a humid zone, so that's a good start. Insulating with walls with XPS or EPS foam will help too, it'll keep the moist air further away from the cold wall.
I can't say if an air exchanger would help or not. If it's including an A/C system, it could help (as the A/C is a dehumidifier) but otherwise I have a hunch you'd just be pumping more humid air into the space.
Products like drylock don't do a whole lot. They're not strong enough to prevent hydrostatic water pressure (high water table) and do nothing to prevent condensation. They're a bit of a gimmick.
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The advice to remove water before it comes in is sound but if you run a dehumidifier and the rest of the house isn't extremely tight, it seems unlikely that insulating the attic is going to cause a major difference overall.
It really isn't a problem that your land is flat. The grade required is very sight because the actual drainage will be to a pipe. If you live on a street with a storm sewer, it will drain to that which will be below your yard regardless of how flat it is.
The biggest source of water is likely your roof, not the neighbors pool. If you have old clay drainage, it's probably clogged and cracked and mostly non-functional. And it's probably installed right up against the foundation. This means water comes from the drain spouts, goes into the clay pipe and then (because it is porous) bleeds out around your foundation because it has no where else to go.
Contemporary drainage is completely different. It takes the water direct from the downspout to non-porous PVC and takes it away from the house, typically to a storm sewer. You can also add drains to other areas of the yard where water pools. All of this may be subject to code.