Does anyone have any tips or suggestions on how I can improve the speed of my spoken Chinese without losing clarity or making lots of mistakes (which is my current problem when speaking too fast)?
The shape of your mouth and the muscles you need to speak effect the way that you speak another language. So you will naturally find yourself tripping over certain sounds or combinations of words.
The only way you can improve this is by practicing.
I have been casually teaching overseas students for over 10 years and the best way to do this is by doing two things: Read at a suitable level so you can read fast enough and read aloud so you are training your mouth and getting feedback via your ears.
I started by practicing with childrens' books, you will find your mouth starts to hurt after a short while of reading if you are doing it constantly. Practice daily then you will notice your speaking will also improve.
Songs are a good way to train with, however this is usually via memorisation so it may improve the speed of your spoken Chinese, but not your conversational Chinese, that is, it's not going to teach you to speak better. Because speaking in a conversation requires you to consider a response and to have the skill to present that response with the right vocabulary and grammar.
Practise reciting news broadcasts (that's the speed you should aim for, not rap songs (Jay Chou is pretty notorious in that regard)) and speak more Chinese. You will gradually adapt to the speed of your regular conversation partners (at least those who can tolerate a learning speaker)
If you really want to practice with Jay's songs, try a ballad like 东风破