I think the most important thing for AAT is to help you establish the feeling or the sense of the language instead of bits and pieces of single words, or a flow. the liasons, and connections just do that part. I agree with you that no many local people here speak the same way as what is demonstrated in the book. however, if you keep practising liasons and connections, you will find that your english will not sound that choppy anymore.
And don't forget the intonation (emphasis), lots of chinese still use hong kong or mandarin intonation when they speak english.
just my 2 cents, and i will definitely work on it when I have some extra time.
i will say keep on doing it and do't quit, maybe focus on a few areas for an extended period of time before you move on to the next topic or lesson. record it your exercise and cmopare it with the orginal, if you can't tell the difference, it means you still got a lot to do here. don't run before you can walk.
good luck