Yes, I would definitely use a different index. You need an index which has both columns. Try one of these. The second should be faster, but you are only allowed to have one clustered index per table. Use the clustered option on the index you use the most. Do one of these from mysql command line interface.
in reply to DBI speed up needed on MySQL
CREATE INDEX exp_update ON campaigns ( mw_export, last_update )
CREATE clustered INDEX cl_exp_update ON campaigns ( mw_export, last_up