I think you will find that your BSEE curriculum will be very different from your AA training. It will be much more analytical (much more mathematical) with a more thorough understanding of physics and electrical theory. In addition, you won't just "plug numbers into formulas." You will learn how the formulas were derived and you might be required to show that YOU can derive them. (Don't let this concept worry you. You will be given enough math training so that you will be able to perform these operations.) Such exercises bring real insight into the meaning of functions and equations.
Different people may do things differently, but designing with transistors (or anything else) is not haphazard. Certain things must be taken into consideration and results of certain types must be obtained. Design is an open-ended problem, meaning that there typically aren't fixed recipes for doing things. Engineering wouldn't be much fun if there were.
An EE curriculum will expect you to analyze a whole lot of circuits. Most of these problems are contrived so that the solutions are relatively straighforward. These sorts of exercises will get your feet wet and will give you a sense of what's important and how to attack various types of circuit problems.
You can probably expect to do some real, if rudimentary, design in something like a senior design seminar. Solving real problems like this in an environment where there are professors available for coaching and fellow student who can offer their ideas is very helpful.
BUT, when you get out into the real world and get real design assignments, you'll probably be lost for a while. Your real design training will be under the tutelage of an experienced engineer. This will happen if you're lucky. If you're not lucky, you'll have to absorb these skills much more slowly on your own. But there is still hope: transistor data sheets aren't really as daunting as they might seem right now.
Don't be so sure. You don't learn how to analyze transistor circuits without learning about transistor parameters.
Both of these statements are wrong.
When you analyze a circuit with pencil and paper, you can only consider a few parameters at a time. Spice-type programs can consider many parameters at once. But Spice programs don't give you the insight that's possible with pencil and paper calculations, and they often give erroneous results. The problem with Spice programs is that they are strictly numerical and it is relatively easy to introduce errors and instability into numerical calculations. (Computers are not the be-all and end-all of problem solving.) This problem can lead to erroneous answers or even the failure to converge to a solution.
Don't expect that you should understand this stuff BEFORE you go to school. The concepts needed to understand harmonics and frequency multiplication are simple. You will be taught all about this sort of thing.
Don't be so quick to judge. Take the BSEE courses and I think you will change your mind.