A computerized payroll system that has left thousands of Los Angeles public school and community college employees improperly paid will be adopted by state government next year despite its many problems. The payroll system would help the California controller's office generate paychecks for more than 290,000 state employees. The office administers an $18 billion annual payroll that includes the state civil service, judges, elected officials and more than 59,000 California State University system employees. California officials say they are working to avoid problems that have shortchanged faculty, cafeteria cooks and other workers in the Los Angeles Unified School District and Los Angeles Community College District.