The very fact that you have said that you have always been bothered by this suggests that something inside you is niggling at you to learn the 'best' way you can, especially if you love to play the piano. This could be your intuition speaking to you and if it is, then you should perhaps listen to that little voice because it could, and will, help you in the long term.
If you have been learning in a piano class it is very easy to understand why you got fed up with the beginner songs and wished to create some 'real' music.
I have to say that if you really do like the idea of being able to play the piano and to be able to have the ability to express yourself in the best way you are able, then it goes without saying that private lessons are going to be the best move for you.
One of the problems is having the desire to run before you can walk, especially for the adult beginner but you know, getting a solid technical foundation under your belt is really quite important in order to allow inner creativity outer expression. In reality, there are no quick routes, no short cuts.
One reason why we learn certain fingerings for scales is that we learn to recognise a pattern of fingering which we will remember and utilise in other pieces. So every piece and every scale gradually builds up a whole set of patterns which will come in useful later on. It is actually much harder to have to relearn fingerings for recurring patterns which could have been learnt in one swoop.
I agree, it does seem a long process but remember, the feeling you have while practising 'now' is the same feeling you will have in the 'now' of next year. Have you ever seen Ground Hog Day? This is similar to the 'now' I'm talking about. Today, in this 'now', you may not be able to play something, but if you keep persisting and practising, you will be playing a great deal in the 'now' of a few years time. If you use method in your practice today, you will reap the results at some point in the future and at the same time, will feel good and more confident that you are learning in a way that is going to benefit you in the future.