It is true, a Boston-Dallas route via New York (New Haven), St. Louis (Pennsylvania) to Dallas (MP-T&P) would probably be the fastest, but it would require changing stations in New York and two changes of train en route. There are a variety of routes. One involving only one change of train en route would be New York Central’s “New England States” from Boston to Chicago, then Santa Fe’s “Texas Chief” to Dallas. Baltimore and Ohio and Missouri Pacific periodically did offer through sleeping car or slumbercoach service between Baltimore/Washington and either Fort Worth or San Antonio, depending on the year.
I don’t have a 1963 timetables, but from October 29, 1961, the New York Central timetable indicates the coach fare from Boston to Chicago to be $43.50, and $29.33 on the Santa Fe between Chicago and Dallas. Roundtrip coach it was $76.68 on NYC Boston-Chicago, and $52.80 on Santa Fe between Chicago and Dallas. Sorry, I don’t have any MP timetable that gives fares, but the NYC-ATSF routing would probably be about the same. Since you asked for the “cheapest way,” these are coach fares…sleepers are more.
Sample itineraries are (from a March 1963 Official Guide):
NYC-ATSF (1 change of train)
Dp Boston 315 PM (ET) NYC train 27, the New England States
Ar Chicago 915 AM (CT) NYC train 27, the New England States
transfer from La Salle Street Station to Dearborn Station
Dp Chicago 600 PM (CT) ATSF train 15, the Texas Chief
Ar Dallas 155 PM+2 (CT) ATSF train 115, the Texas Chief (train 115 was a section of this train from Gainesville, TX to Dallas)
NYC-CRI&P (2 changes of trains)
Dp Boston 315 PM (ET) NYC train 27, the New England States
Ar Chicago 915 AM (CT) NYC train 27, the New England States
transfer could also be made at Englewood, IL, but both NYC and CRI&P used La Salle Street in C