If someone wants to avoid using the shell, or does not have shell access, I have had good experiences using a GUI like SQLyog Community. With that you can connect directly to your current database host as well as the new one (Dreamhost). Then there is an option to “Copy Tables to Different Host/Database.” You can select all tables or a subset of them.
There is an option to “DROP if table exists in target” which you may want to use, but be careful and double-check of course. :]
This should not run into issues with database sizes. As far as I know it is using individual SQL commands to copy.