Universities that require students to apply directly to a specific major or college are what you want to avoid - exploratory majors may reduce choices (at Purdue or Uiuc for instance, it’s very hard to get into engineering if you are in Exploratory).
Case Western, Brandeis, UDenver, Tulane may be better options, as well as all LACs.
Note that engineering is going to be more prescriptive than other majors, with specific sequences to follow. At many smaller universities, the first year business classes are cross referenced with other majors’ (freshman seminar, math, Economics, Foreign language, science) which allows easy switching.