If anyone is interested in the progress of this problem. Here's an
update:

I contacted Culligan, and they came out to do a water test. The
gentleman who did the test is very confident we are suffering from
"colloidal clay" (which is pretty much liquified clay). Also, the
water is a bit acidic (5.8) and hard (3gpg). He is taking a sample to
their lab for confirmation, but he's pretty sure this is what it is.

He said that the causes for this point, most likely, to a problem with
how the well was drilled. Either the casing down the well is cracked,
or the well was dug a bit larger than the casing, but either way ground
water is coming down and gettinng into the well. He said that the
water is prematurely getting in the well from above somehow because
these results (except hardness) don't usually occur naturally at this
depth of well.

The solution is either to install a Reverse Osmosis filtering system or
dig a new well.