The challenge is not many seats available from your origin especially for 4 persons (e.g. connecting via EWR is on a ERJ {smaller aircraft with only 1 unassigned seat at the moment} or connecting via SFO {@7:40am} only has 1 unassigned seat left). And you don't wish to go stand-by with a family of 4 plus if checking bags you want them on the plane you are flying. As someone else mentioned UA codeshares seats on Air China which has a flight departing IAD @ 4:05pm so even if delayed you have 4 hr to make that flight. If possible bring a charged phone or tablet and have it handy on your UA 778. If delayed after leaving gate (ATC hold) - you can use the app inflight to see if they rescheduled you to Air China or as soon as you land call United to advise of misconnect (likely faster than waiting for customer service counter @ IAD after getting off plane).
So, IMO remain on your schedule flight but monitor the two following websites - nothing you can do except write down UA 7609 which is UA/Air China codeshare.
www.faa.gov (check delays or ATC issues at both IAD, ATL & DCA)
Check UAL for your flight including inbound aircraft (which is UA 1155 scheduled departure at 7am EDT)
We all wish you and your family luck as Mother Nature has been quick erratic recently.