They will be very rare your first day till you get closer to Kansas area, then you will likely find some that offer electrical but no water. Depending on your travel habits and distance you drive per day, you could be in decent weather 2nd day, just don't count on it because being January anything can happen.
Having said that we have stayed overnight in Stromsburg, NE in a small municipal c\g right on 81. Water and sewer were not available but electrical was so that worked just fine...there are other places like that along the way...but we did that in a motorhome...easier than with a trailer, since heat is always on.
Personally, with a trailer, I would just count on staying in a hotel the first night, and then play it by ear after that.
2016 Chevy Silverado 2500 Duramax
2016 Rockwood 8289WS, Diamond Pkg.