These subject are so diverse. Most engineering students have no clue what real engineering is until like their senior year. Not having math background at some schools would be a big no. Most schools for engineering want you to start math at Calc 1 so having precalc is important or Calc 1 now.
It is usually not easy to switch “into” engineering at most colleges. Michigan is the one I am familiar with that it is not that hard to do. Maybe find another school that switching from LSA into engineering would not be difficult. This way you can try out classes and see how it goes.