water coming up from under cement foundation
OK, the water must be coming from somewhere:


Water from a leak. If you've been checking your meter and don't think you have any leaks inside that's good, but it's possible there is a crack or break in service line upstream of your meter. This might actually explain reduced water pressure in the house. If you suspect this is the problem I would call your water company and have them take a look.
Rainwater. If the water appears only during and shortly after a storm, then you probably have a drainage problem. Make sure all gutters are directed away from the house. You might also look at grading near the house to make sure water will flow away from the house.  Is the porch at the bottom of a slope towards the house?
Groundwater. I would be a little surprised if this were groundwater welling up, but it's not impossible. Have you had a very wet spring that would've raised the water table level? Or done any serious landscaping / excavation / compacting in your area?
Septic system. An unhappy septic system can cause water to well up, although probably not so close to the house or clear / odorless. Do you have a septic system?


(I wouldn't beat yourself up too much about not sealing that gap. If water is pooling from below I doubt a little sealant would hold it off for long.)