I am trying to figure out the appropriate statistical procedures to analyze data....
... essentially, every patient will be his/her own control in this procedure, and we're comparing each person's slope before and after implementing their new approach to clinical care.
You've been given some solid advice regarding statistics for analyzing if there has been a significant change in the slope pre- versus post-change. Depending on the type of data and how it is measured things like time-series analysis, statistical process control, etc.
MAY be some options. Statistical analysis of such repeated measures from the same subject can be problematic (e.g. due to issues with autocorrelation). Also, assuming multiple clients, you face the issue of doing individual analyses on each client (time consuming) or aggregating the group data (blechhh!, speaking as a behavior analyst). Note that these methods, best case scenario, would only demonstrate that there was a significant change in the slope. In the absence of any experimental control, you should not and cannot conclude that the change in client data is functionally related to any specific changes that you made in your way of doing things.
There are, however, several research design methods that don't involve stats where you use individual subjects as their own controls (I currently teach a whole graduate class on these "single case design" methods). If you've already implemented the changes across the board with all clients, then you really are limited to only one such experimental design at this point- a reversal/withdrawal design where you would go back to the conditions in the clinic before you made the changes until the data stabilizes, and the re-implement the changes. If the data returns to baseline levels when you revert to pre-change conditions and goes back to post change levels after you re-implement the change, you can conclude that there is a functional relationship between the changes you made and the change in the client's performance. There are ethical issues with using such a design in an applied setting, including doing something that would lead to client regression. You could potentially do this with just a few of the clients. Otherwise, you are pretty limited in what you could use, post-hoc, for experimental designs. If you did not implement the changes for each client at the same time, you might have accidentally stumbled upon a multiple baseline across subjects design. Going forward, if the agency is willing to use the older pre-change procedures for awhile with some new clients, you might be able to do non-concurrent multiple baseline (but with no overlap between the baselines in the different groups, it would be a pretty weak design).
TL/DR- there may be some ways to show that the data changed significantly after the change in procedures. However, this would not show that the change in the procedures caused the change in the data. There are some single-case design options for showing causation, but having already made the changes in procedures, you're kind of limited.
