Nice start tackling this challenging section.
The sample size estimate seems pretty large for your effect size. Try these steps:
1. Assume for now just simple pre-post (not accounting for changes in control sites).
2. Calculate the sample size for a simple chi-square (Test of proportions). Let's say the baseline screening rate is 10%, and you would like to introduce an intervention that would increase this to at least 20%. And assume equally sample sizes pre and post. Now using a sample size calculator, what do you get?
3. The next step is to account for the sample size inflation that will be necessary to account for the clustering (i.e., the design effect). What is your design effect? This will be driven by how many eligible diabetic patients you think will be visiting the psychiatrist, on average, during the study period. Consider looking at several design effects and sample sizes assuming ICC of 0.02, 0.2 and 0.5...
4. Multiply unclustered sample size by design effect… is it the same as your previous estimate?