NE doesn't have a lower chamber, just a Senate. In most legislative documents NE is left grey for the lower chamber. Also all Senators in NE run without party affiliation in a top-two primary. I assume that you are imputing the party of NE Senators from other sources.
With NE just having one chamber, my plan is to include them in both the lower house and upper house maps I intend to make. I actually was not aware they were called "Senators" but by any other measurement, I don't see why they should be considered either a lower or upper chamber exclusively. The alternative - a big gray blob in the middle of the map - is just too aesthetically displeasing for my tastes.
And yes, their affiliations are drawn from a combination of state party endorsements (virtually every candidate gets endorsed by one or the other) and on-the-record statements by each individual.
South Carolina still has that many Democrats?
In sheer numbers, they only hold 35% of the chamber - not inherently out of line with many other comparably gerrymandered states.
However, they may very well hold districts comprising a majority of the state's geography, since most of the Black Belt counties are sparsely populated and such a large chunk of the state's population live in clusters either in the far north or south of the state that are substantially GOP. Presumably, the number of VRA-protected districts required had to be expanded in size to such a degree that they would encompass a much larger share of land mass than in many other states.
Mississippi is another good example (although perhaps not as extreme): Democrats still hold 40% of the seats there. A combination of a state being a) more rural/less urbanized and having rather homogeneous demographics spread across it, and b) having a very solid floor of Democrats (particularly if they're VRA protected) leads to this phenomenon. Ditto for Louisiana to some degree, and even AR until recently.
Contrast this with states like GA and NC, which have more substantial shares of their populations in a few urbanized clusters + less diverse/homogeneous rural spreads, and that fuels the difference.