Upvote:3
"Do not change your activity" means "do not start doing anything that is forbidden under your current status." Tourism is, however, allowed in E-3 status. Furthermore, there are some statuses where maintenance of the status depends on staying employed, but at the end of the employment there is a grace period within which one must leave the country or file an application to change status. As long as the application is filed before the end of the grace period, it is not necessary to leave the country, and as long as one can be in the country, one can travel.
So yes, you can stop working.