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. See below...
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.
You can find anecdotal evidence on the net indicating both strategies will work and that an 'implicit switch' (i.e., where there's no hard evidence of a status change) avoids a needless exit/re-entry solely to satisfy a formality. So your choice will ultimately depend upon your own appetite for having a clear paper trail.
Note also that we have a lot of questions/answers 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'. This aspect of your question is stated explicitly in the rules.
The Schengen Borders Code is here.