There is nothing explicitly stated in the rules governing the switch from EEA resident to Schengen tourist except that you need to have a passport that allows for visa-free entry. I took a course in Schengen earlier this year and asked the instructor (Elspeth Guild) what she advised clients in this situation...
Wouldn't that mean that I can quickly exit and re-enter the Schengen area to activate my tourist stay?
Yes, this is the recommended strategy. This enables you to cap off your residence visa and start a new stay as a short-term visitor with appropriate documentation at every step.
Also, is there a way to change this purpose of stay without exiting the Schengen area?
This depends upon the host nation and if they have a policy in place for an 'in-country switch'. If they provide the right passport stamps documenting that you did not overstay your residence visa, then fine.
Note also that we have a lot of questions here explaining that time spent on a residence visa is not reckoned against the Schengen 90/180 day counting rule and this will apply to you. Your Schengen clock will activate with the full 90 days regardless if you exit/re-enter or make an 'in-country switch'.