The routings that the OP is looking at (NYC-MIA, MIA-HOU, HOU-SFO, SFO-LAS, LAS-LAX) can be done almost entirely on reasonable o/w fares and, in most cases with great frequencies. SouthWest and JetBlue operate in almost all those markets (or close enough to them) that the total domestic flying budget can be very reasonable. Switch MIA to FLL and it is pretty much entirely a B6/WN routing, maybe needing to do OAK instead of SFO, too. Have the RTW ticket put you into YYZ/YUL/NYC and out from LAX/SFO and go from there.

As for seeing the country, there are certainly parts where driving through is a great thing, but after 1000 miles of corn fields in middle America I'm not sure that it really is worth it. Drive LAS-LAX, HOU-AUS and take the train on the lumbering, ridiculously long YUL-NYC and you'll see plenty. The rest of it you can see from 30K. Yes, you'll be missing things by doing it that way, but I don't think that it is a terrible trade-off.