I've traveled NE Oklahoma to Detroit area in winter at least 50 times in the past 35 years, and some winter trips as far east as Cleveland. I've had "we need to stop and wait this out" weather no more than five trips.
Weather can be clear, sunny and cold, or warmer and wet. "Wet" can be blizzards anywhere along your route, but from St Louis to the west and south freezing rain (ice storms) is more common.
Frequency of winter storms across the plains, where Gulf air has to push into Continental Arctic air masses might be once per 6-8 weeks in the north, to once a week in the south, closer to the Gulf. Duration is a day or two.
Eastern part of your trip is a different problem. You get lake effect snows until the lakes freeze over, some years that's never, and seldom earlier than than January. Problem areas are west of Lake Michigan (SW Michigan, northern Indiana, NE Illinois, a problem if you are going to Chicago before heading south), and from Toledo on east (Ohio, Pennsylvania, New York) for snows off Lake Erie and Lake Ontario. You might see small lake effect snows almost daily as air masses move west to east, picking up heat and moisture over the water. You'll get big lake effect snows as major frontal systems come across, 2-3 week intervals, when heat from the lakes reinforces that from Gulf air masses.
Travel options include going all the way west on I-90 and I-80 to I-35; taking 8-90 to I-55 out of Chicago to St Louis, then crossing on I-70 to I-35, or taking I-44 all the way to OKC; dropping down to I-40 on I-57 from Chicago, then I-40 to OKC; dropping down to I-40 early using I-71 out to Cleveland to Louisville, then I-65 to Nashville.
I mention all these routes, no more than one day's difference in travel time among them, because that flexibility is the reason I'm batting 90% on avoiding bad weather across that part of the country. I check the forecast for the next few days, and choose the route that avoids the problem areas. I've a handful of additional routes using the US Highways and state highways in Kansas, Missouri, Illinois and Indiana, but in winter, with a RV, you are better staying on the Interstates, they get cleared sooner and more often and you'll find services open.
I do not often go the "get south quickly" because the I-40 corridor gets bad winter weather more often than the northern routes. The heat and moisture for the winter storms comes from the Gulf of Mexico, and it reaches Tennessee and Arkansas more often than it reaches Illinois, Iowa and Kansas. The northern routes are more often clear and dry, but also can be bitterly cold when the weather is clear.
There are no "sudden" winter storms across plains and corn belt routes, in the sense of coming up unpredicted. Major air mass movements are needed to produce the storms, predictability is 3-4 days, but timing is not precise, can be 6-12 hours off anywhere along the path. Much like predicting hurricanes.
"Sudden" storms, i.e. for which you have no long term predictions, are more common along the southern corridors: I-40, I-30, I-20. Forecasts are still at least 2-3 days, but a system can fail to push as far north as expected, and bad weather doesn't show up, or it comes further than forecast and a place that was supposed to be clear gets rain, freezing rain, or snow. Also, it is hard to predict form of precipitation in these air mixes, a warm front supposed to produce drizzle can get cold air under it and produce unexpected sleet.
East of Toledo, I can't help you with alternate routes, I don't have the travel experience there. I don't even pay much attention to the weather forecasts, since I don't have anybody living there.
Tom Test
Itasca Spirit 29B
