I live in a tri-level house so the entry way, half bath, laundry room and family room are all located on a concrete slab. Well we just ripped up the floors in these areas as they were extremely old and dingy. The slab in the laundry room and part of the bathroom (connected) appear to be pretty damp. My question is, what can I do, if anything, to fix this problem before I install the new flooring? I do have a sump pump in the basement and everything appears to be fine down there. Can I just put down a sealer like RadonSeal, place a humidifier in the room, or do I need to do something bigger? Any input would be appreciated.

I think you meant dehumidifier. It is important to determine what is causing the dampness. You can not seal already damp concrete. Air out the area for a day using box fans to circulate the air and use the DEhumidifer if you have one. After its dry cut a small square of plastic bag or sheet plastic and tape to the floor with duct tape around all edges. Wait a day and then peel back the plastic , is it wet? If so do not under any circumstanes seal or move forward with the new flooring

Yes, I did mean dehumidifier. Thanks for making sure though. Ok, I'll try this test, but what if the plastic is wet? What do I do then? Call in an expert?

How old is the house?
Depending on the water table where you are, and the type of soil, and the drainage originally installed, you could have a major renovation all around the foundation. But it means excavating down below the footings, installing new drainage (new methods are much better than years ago), and waterproofing the exterior walls with a sealant, a membrane that looks a little like small bubble wrap, and possibly foam insulation if required where you live (although studies are showing that it's useless outside, where bugs channel all through it).
I'd go to the town and get the building plans on file to see also how the slab was done - any insulation or vapor barrier under it. If not, you are facing a tough situation and might just either use tinted concrete sealer or a fake laminate floating click floor, cheap but it's not worth ruining another floor.

The house was built in 1985. Not liking the sound of a major excavation. Hope it doesn't come to that. I am wondering if maybe they did do a vapor barrier or something with original construction but it got a hole or something under this part. My garage is also connected, is about foot lower than the damp area, but there is no traces of moisture. Or is it possible that the vinyl that was once there just didn't allow for the concrete to breathe and the moisture just accumulated over time? If you can't tell I may have no idea what I'm talking about. I'm just trying to figure this out.

Don't want to add to your concerns but are there any water pipes underneath those floors?

Also, how much of this house is below ground? When you say basement with sump pump, is that built into the slope and the other rooms on the slab open out at ground level? And the 'basement' area is dry? That could be dry because the heating system keeps it dry, and the floor isn't covered. The garage could be dry because it's ventilated. As stated, there could be a leak under the slab.
I don't think a house built in 85 should necessarily need a major excavation around the footings, but who knows how they did the drainage.
Maybe do the plastic test (about 18" square) in several areas including the garage.

Personally I'm not sure I would put down any kind of flooring on a slab needing a sump pump. I'd use tinted sealer and synthetic rugs that can be aired out and replaced. It isn't just dampness that accumulates. Bugs like to live under any kind of flooring where there's moisture.