ask your support techs , they would be able to switch the charset of your database using a single command in the backend . Make sure you tell them which char set you want to switch the db to
AFAIK this is not always possible
it depends on how mysql is organized there - if there is separate config for each account for example.
if there is one common for all accounts i doubt they would like to change charset to something exotic because of just one user
then user needs to edit his php scripts and set proper charset in that part of code responsible for mysql connection.
there are also some scripts that would rewrite sql file, fix charset and then u can upload it to the new server