You got it. Problem is the turncoats. Many politicians who know they can't get elected as a democrat in their districts are running as RINOs and diluting the Republican party. That said, do the true Republicans really have a majority in congress? I would say no.
The GOP House members have 18 or so RINOs who unless they are given their pork will go with the other side; in the Senate, there are about six. The trouble with the Senate is that it takes at least 60 to do anything and that's a troubling new thing. It used to be filibusters were saved for really momentous issues, like civil rights, and the huge budget fights. Now, everything is filibuster-albe (if that's a word). If I were in the Senate, I think I'd restrict the filibuster...but it takes 2/3 to change Senate rules and I'm not sure we want to take away all obstructions to bad laws. Still, compare the Dems at their zenith: more than 100 seats in the house and more than 60 in the Senate. By comparison, the "Republican majority" has been razor thin.