You can continue to stay in Japan without changing the status of residence.
For example, even if foreign national with the status of Engineer/Specialist in Humanities/International Services does not change his/her status of residence to Spouse or Child of Japanese National after he/she marries Japanese national, penalty would not be applied. However, in that case, it is recommended to change the status of residence to Spouse or Child of Japanese National.
This is because those who has family related status of residence such as Spouse or Child of Japanese National can engage in any activities without regulations on the contents of job. Therefore, changing the status of residence is recommended unless you have special reasons.
On the other hand, it might be better to keep the current status of residence until it expires if the remaining staying period is 2 or 3 years or longer. This is because, in many cases, wife or husband of Japanese is given only 1 year staying period when changing his/her status of residence to Spouse or Child of Japanese National.
By the way, if you would like to engage in the activities that are not permitted to your current status, Permission to engage in activities other than permitted under the status of residence is sometimes necessary.