Yes, in 2 QB leagues they will generally go off the board much quicker. The size of the league also has a big influence on it as well. If it's an 8 or 10 team league, then there will still be solid #2 QBs available later in the draft. However, if it's 12 teams or more then you start getting into the lower tier starting QBs that some teams are going to get stuck with as a #2.
Depending on where you pick, you could go with either QB or RB. As I said, league size plays a factor. If you can get Rodgers, Brady or Brees take them. If it's between Foster/Rice and Stafford then it's closer.
When picking your 2nd QB it's just something that you have to make a judgement call on for when to pick them depending on how the draft is going